I'm going to leave more qualified people to give a full answer, but I can comment on one aspect of this question. In the 2^nd century, an author called Lucian of Samosata wrote a book called "True Stories," about a group of adventurers who get caught up in a celestial war between the moon and the sun, before returning to Earth and going on further escapades within a massive whale. It was actually written as a parody of common legends at the time; in the introduction Lucian mocks his contemporaries for believing in such obviously false stories such as the Odyssey, those of Ctesias the Cnidian (author of a number of histories regarding India and Persia), and those of Iambulus (who wrote legends of islands in the southern Indian Ocean).

The Iliad and the Odyssey were highly regarded by the Greeks and Romans as great works of literature, but I'm unclear on the breadth of their acceptance. While many late Republican and Imperial Romans seemed to take an agnostic view of the universe, even a figure as late as Augustus saw the use in them for propaganda (the Aeneid was composed by Virgil at the behest of Augustus to give the emperor's family some legitimacy; supposedly the Julian family was descended from one of Aneas' sons). My theory is that the common people; the unemployed, the farmers, and the laborers believed in the stories, but the more educated merely saw them as well-written pieces of fiction (similar to how we see Dickens today), to be studied in school but viewed with a skeptical eye.

This is a rather difficult question to answer. Who were the Greeks and the Romans? (Sorry, I know, a very typical AskHistorian opening but this needs to be said!) Take the Romans. Are we talking about the beliefs of the senatorial elite? The priests? The peasants? The slaves? People only in the city of Rome? Or Italians? Africans? Gauls? Syrians?

There's not really right answer to this; you could walk from one end of the Roman empire to the other, and face thousands of diffirent divinities and myths, and different people engaging with them in different ways. I would say that yes; the great majority of people were definitely extremely superstitious and religious, and their life was defined by trying to live in an equilibrium with unseen forces of the universe through rituals, curses, complicated rules, and so on. And they needed some methods to explain the world as they saw it. But the question about myths is a bit more difficult.

Firstly, the ancients didn't have a category of things that we in modern day would call "myths" or "religion" or "magic". Our modern Western concepts of these things cannot be applied to the worldview of ancient Greeks and Romans. (Tip: everything written by John Scheid about this topic is solid.) Yes, there are definitely stories featuring mythical beings that the great majority of ancients would consider fairytales. Think about Plato, and how he uses myths about gods and fantastical creatures as a rhetorical device to get a point across in his philosophical dialogues, although he himself doesn't believe in human-like gods characteristic to the Greek pantheon.

But yes, lot of ancient history can be considered 'mythohistorical'. I'm taking Livy as an example, because I'm most familiar with him. He is definitely skeptical about fantastic events and supernatural forces, that often feature in the historical sources known to him. His attitude is basically "I'm just telling ya what everyone's saying, and I don't really buy it, but you can make the decision for yourself" (See Levene (1993): 16-37 on Livy and skepticism). But at the same time, there are lot of characters and events in Livy what we would consider myth, but what he considers historical. To be fair, some of the events he's writing about happened 700 years ago, and he himself moans how difficult it is to find the 'true' account of events so ancient. But he for example believes that Aeneas was a historical character, or that myth of Lucretia was a historical fact, or presents a historical narrative for how 'Roman religion was born', but obviously in truth we can't account the birth of Roman religion to certain individuals or events.

Whether most people believed into Iliad, Odyssey, or Aeneid is a bit difficult to say. Is it too annoying to just say some of them would, some of them didn't? It's quite unlikely that the majority of, say, Romans during the high empire would have been all that familiar with the Aeneid, or Homer; they are works characteristic to the elite culture of literature and education, which also of course formed the most skeptical part of the population. There might have been oral versions circulating around the empire, versions which might have been quite different from the forms we know today. The Aeneid and the mythical origins of Julio-Claudian emperors was very much a feature of especially Augustan propaganda; so Augustus himself surely couldn't have believed most of the stuff in the Aeneid, as he consciously made it up and used it for his own PR purposes. But I don't see why most Romans wouldn't have taken the propaganda at face-value, seeing that it was an empire of rather uneducated and illiterate people. There is probably someone else in rAskHistorians who can talk more about the Aeneid and it's reception throughout the Empire's history, it's not really my specialty.

I'll take the pagan myths of the after-life as a quick last example, as this is my research field. I'm working on the interplay between Roman philosophy, death and dying, and Latin epigraphy. (Valerie Hope Roman Death and Catherine Edwards Death in Ancient Rome are great introductory books on this). Did the Romans believe in Tartarus, Cerberus, River Styx, ferryman Charon, and all that? The real answer is probably: yes and no. The Romans simply did not have consensus on how to explain after-life, or only one accepted version. Unfortunately we don't really have any sources on what the common people thought about after life; what we know comes from mainly elite literature and fancy and expensive funerary stones. The elite philosophical movements, such as Stoicism and Epicureanism, completely denied the existence of after-life; but not everyone was a philosopher. There are a lot of funerary inscriptions that feature elements from the pagan myths about death, and the Romans definitely had a cult of attending to their death; such as specific festivals when they went to pour libitations, feast on their dead ones' graves and such. But of course, to some this could be just a tradition like Christmas, which wasn't necessarily about the dead getting their share as much as a nice family day out. There are lots of funerary verses, that clearly use the pagan myths only as a poetic device, to paint a pitiable picture of the deceased's current miserable state to get a tear out of the passer-by. One funerary stone springs to mind, which features separate verses for the dead husband and wife. The husband's verse is very much in line with Epicurean philosophy, saying that he's in a happy state of non-existence; but the wife's epitaph is very much about the ground weighing heavily on her while she wanders in the gloomy fields of Hades. It is possible that the wife and husband had different outlooks, and they wanted their own ones commemorated on the stone; but the more likely conclusion seems that the both belief systems are used as 'empty' poetic conventions of Roman funerary monuments.

To finish off, here's one of my favorite funerary verses discussing afterlife, a marble slab from Rome (prob. 3rd century AD), of the deceased lady Cerellia Fortunata:

Traveller, do not pass by my epitaph,
But stop, listen and, when you have learned, carry on.
There is no boat in Hades, no ferryman Charon,
no Aeacus, holder of the keys, or any dog called Cerberus.
Allow us [who] have died down here into nothing
but bones and ashes have turned.
I have told you as it is. Now go forth, traveller,
So that after death you might not think, me to be over talkative.

On this tomb no ointments or garlands
As offerings lay, for this just a stone.
Light no fire; it too is a wasted investment.
If you want to give me something, give it in [my] lifetime.
Wine poured on ash won't quench the departed's thirst,
all you'll make is mud, and that'll be me.
Instead, having thrown a handful of soil on [the graves], say:
"What I was, when I did not exist, I have again become."

For the most part the answer to your question is yes.

I am going to go mostly into Roman history here because that is my field of interest but much of what is said can be generalized toward the Greek. The Romans were extremely religious, with religious ceremonies surrounding much of there lives. Each family would have a small alter or shrine called lares where they would worship the family gods daily. In public there were temples where public services were held and offerings were made.

A large part of the issue Romans had with the Christians for example was the requirement that every person in the empire pray to the gods. The Jewish peoples at many times had legal exemption from this "oral sacrifice" but the Christians did not. Many magistrates with no particular hate for the Christians gave many opportunities for Christians to preform the oral sacrifices required of them but to the Christians in question it went against the "no other gods but me" clause. This was so unacceptable to the Romans that they were sentences of death by crucifixion. It was that important to them.

Cicero's On the Nature of the Gods says some interesting things on atheism. Its worth a read by itself but to summarize... Philosophers have not proven the existence of the gods, and uses the existence those that do not believe, though Cicero is careful to not mention any Romans among the list of atheists, as a proof that it is not "natural to believe that the gods exists". There are many modern scholars that believe Cicero was indeed an atheist but even here you can see the care taken to distance himself, and any good law abiding Roman, from atheism.

On the stories of Homer and common tales of the gods on the other hand... it is hard to say what people believed until I finish my TARDIS. I would hazard to say that the tails provided moral teachings, similar to the tales that many modern religions give us. Were they believed literally? Many people believe religious fables literally today, many believe in the morals as the basis for there telling, many don't believe at all. I would say it was probably the same then.

Just to throw some extra comments into the fray... There have been a couple of reasonably recent discussions on similar topics:

How do historians feel about drawing conclusions from oral traditions? how accurate are they deemed, professionally speaking?

and Before Virgil and the Aeneid, did Romans believe Aeneas founded the Roman race, or was that idea constructed by Virgil as propaganda?

They're not an exact match to your question. But definitely help give you some nice contextual stuff around the subject.

Now I do have to declare a bias that is probably self-evident from the links above. I think it is important to make a distinction between the Iliad and Odyssey, the Greek epics accredited to Homer and the Aeneid written by Virgil (Like /u/Eshtan said). This distinction is not me just trying to pick up semantics, but had important implications for a historical perspective on your question.

The Greek tradition came from an oral/spoken word culture. The Illiad and the Odyessy evolved and came from a tradition that was simultaneously not solid (i.e. written down) but at the same time had very much 'fixed' forms and ways of telling. The person reciting the poem would have had to stick to the familiar story that the audience knew. Not much room for improv.! Now, whether we can say that the Greeks actually believed this to by true in the same way sense some Christians with the bible is more problematic. Like /u/Vir_Brevis) points out, a TARDIS would be pretty handy at this point. The Bible, the Iliad and the Odyssey came from different places geographically and culturally. But we do have historical textual evidence that the story of the Iliad and the Odyssey were regarded as at least in a basic form, as a remembrance of actual events that took place.

Herodotus and his famous work 'The Histories.' Structurally, Herodotus uses many of the familiar forms of presenting his work that his readers/listeners would have recognised from the epic poems. But most significantly for us, Herodotus explicitly lays out his understanding of the origin of hostilities between East and West (i.e. the Persians and the Greeks). Herodotus attributes this to the abduction of Io and then Europa;

This is the display of the inquiry of Herodotus of Halicarnassus, so that things done by man not be forgotten in time, and that great and marvelous deeds, some displayed by the Hellenes, some by the barbarians, not lose their glory, including among others what was the cause of their waging war on each other. [...] On the fifth or sixth day after their arrival, when their wares were almost all sold, many women came to the shore and among them especially the daughter of the king, whose name was Io (according to Persians and Greeks alike), the daughter of Inachus. As these stood about the stern of the ship bargaining for the wares they liked, the Phoenicians incited one another to set upon them. Most of the women escaped: Io and others were seized and thrown into the ship, which then sailed away for Egypt. In this way, the Persians say (and not as the Greeks), was how Io came to Egypt, and this, according to them, was the first wrong that was done. Next, according to their story, some Greeks (they cannot say who) landed at Tyre in Phoenicia and carried off the king's daughter Europa. These Greeks must, I suppose, have been Cretans. So far, then, the account between them was balanced. But after this (they say), it was the Greeks who were guilty of the second wrong.

^Herodotus, ^with ^English ^translation ^by ^A. ^D. ^Godley. ^Cambridge. ^Harvard ^University ^Press. ^1920. 1.1.0-1.2.1

So, in Herodotus we have evidence to suggest that people may well have believed the story/myth of Io and Europa, but Herodotus gives them, to our eyes, a rational and logical explanation. There is not mention of gods and supernatural happenings, despite the fact that these are very much a feature of the original myths (Io and Europa both being victims of the irresistible charms of Zeus).

So I think we can tentatively chalk that one up as some evidence that at the time of Herodotus, considering his audience and patronage of the upper-classes of Athens, they did not have a belief in the 'literal' truth of some of their mythological past. But that does not mean that the stories were therefore 'untrue.' We should not treat them as mutually exclusive.

So moving on to the Illiad. As has been mentioned, this one is slightly easier as it can be put firmly on the doorstep of Virgil. e wrote it (incredibly slowly). But write it he did. It explicitly links the young and self-proclaimed princeps Octavian/Octavius/Augustus with a neat founding myth of the Roman people and, importantly, the mythological lineage of Caesar. The Romans were keen to draw links to the Greeks and the tradition of the Greek founding myths as they regarded the Greeks as a sort of intellectual powerhouse. If you can show that you are in fact a descendent of the Trojans then you can claim a greater portion of cultural capital. The alternative, horror of horrors, would have been seen as barbarians. Now, we know that the Romans were aware of the more 'mixed' make-up of their people. This is something that is dealt with in the Illiad. There is a necessary intermarriage of peoples that will eventually result in the birth of Romulus and Remus and the more well-known parts of the Roman foundation story.

Now, we should be clear, there is no real definite answer to this, and it is fine that this is the case. In an excellent edition of the BBC radio program 'In Our Time', the academics, including the Classics juggernaut that is Prof. Mary Beard (fabulous human being), get into the intricacies and the controversies of the Roman understanding of their origins.

As Melvyn Bragg (the presenter) highlights in his comments on the episode

Aeneas got short shrift this morning, I'm afraid. There was an attempt to relate Aeneas and Ascanius and their flight from Troy to the small town of Alba Longa near Rome, which Ascanius was supposed to have become first king of, and one of his descendants was the progenitor of the mother of Romulus and Remus, thereby giving Rome a connection way back to the mythical Homeric past which so many European states, civilisations, principalities and even countries aspire to.

The discussion looks at whether the story has any truth in it. A healthy amount of scepticism is exercised. What interests me is the fact that it had such a lasting influence in Western philosophy and eventually conceptions of nationhood.

So, to be critical, the Christian comparison is a tempting one to make, but the match is imperfect. The people in Antiquity had a different relationship with their gods than the Abrahamic faiths did or do. It may be that we could compare say the Roman's with Romulus and Remus, with the Christian understanding of Adam and Eve. They were there at the beginning, played an important role in story the foundation of Rome and mankind respectively. They could have existed in a historical sense. For instance we know from genetic analysis that most humans ultimately come from a very small gene pool ([See discussions about mitochondrial Eve])(https://en.wikipedia.org/wiki/Mitochondrial_Eve). But not in the same way that a literal reading of the story would indicate. It's also fair to say some parts of the bible are historically attested. For instance Herod King of Judea (lots of archaeological as well and literary and numismatics) and Jesus (who is mentioned by Josephus in his work Antiquities). We know the Greek and Romans did 'believe' in the gods, but they were also not fully prepared to accept a supernatural explanation for all the events that took place, as shown by Herodotus in his explanations of Io and Europa. Similarly, in works like the Aeneid, Virgil drew on traditions and myths that were similarly vague in their origins and resulted in a inconclusive progeny. Despite the fact that these works were written down be careful in conflating them with a religion of a book, like Christianity. It is difficult to say that they definitely believed a literal account, because there was no literal account in a written form for them to believe.

If I have to get off the fence I would give a short answer of no; the Greeks and Romans did not believe their myths were literally true. A longer answer would be that the thing is a whole is not that easy. We would need a TARDIS and a bomb-proof poll of people's belief system to know what they really felt, and even this would have differed from era to ear, province to province and individual to individual.

Semi-followup question: I don't think anyone has mentioned Euhemerus of Messene yet in this thread. According to Wikipedia Euhemerus was a Greek philosopher and mythographer writing in the 4th century BC; he didn't believe in the literal truth of myths, but argued that myths are exaggerations and distortions of historical events; this view is still (sometimes) called euhemerism.

Is there any evidence that this school of thought had a significant following at the time? I've been hunting around and can find very little in the way of primary texts; I get the impression that much of his longevity is due to early Christian writers pointing him out as evidence that "see, they admit their gods were false"!

I can answer somewhat for Rome. Like later analysis of the Bible by Christian authors, the Romans had differing views as to how much of their myths was 'real', how much was figurative, and how much was a misinterpretation.

For example, in the myth of Romulus and Remus, Roman authors struggled to explain how a wolf raised two babies. One theory proposed that "lupa" (wolf) was actually a form of "lupare" (to prostitute) - thus, the twins were actually nurtured by a local sex worker.

There was also the matter of Aeneas founding Rome centuries before Romulus. The solution was to have Aeneas found Alba Longa, a Roman predecessor state that Romulus was born in.

To answer the original question: Roman authors certainly recognized that some myths were contradictory and confusing, but they still maintained their overall authenticity.

Source: SPQR my Mary Beard

I will not comment on all the aspects of your question, but the Iliad & the Odyssey can basically serve as an extended metaphor for ancient Greece.
The Iliad was an semi-fictionalized account of the Trojan War, as many of us know. The war actually happened, and Troy was actually destroyed, but the whole love affair/mingling of the gods was rather just to add epic proportions to it (truly, the war was most likely fought on account of Troy being in a prime trading location in the Mediterranean). The Iliad was all about showing how smart & strong the Greeks were, almost like an ancient American Sniper, or a similar film promoting patriotism and (please correct me if this term is improper in its use here) nationalism. Think of this story like Homer saying "Hey guys, remember when we were #1? Good times."
The Odyssey, meanwhile, was rather a look at the failing Greek civilization. Think of it as how Greece, like Odysseus, was lost, and struggling to return to its former glory. It's an optimistic story by any regard, as eventually, after 7 years Odysseus makes it home and slays his wife's suitors to regain his spot as great ruler.

Now, to tie this into your question, I have mostly observed a lot of the same trends as /u/Eshtan. In the same way that modern literature and entertainment is viewed as propaganda and loaded fiction, these pieces weren't, for the most part, considered honest-to-god(s) accounts, rather elegant fiction.