This is a huge question but I am going to give an answer based on two philosophical discussions: Aristotle's, and the Stoics'. (If time and space permitted, we could go through other sources too, such as Empedocles.)

Both Aristotle and the Stoics see large-scale changes in nature as cyclical. This was a motif in ancient Greek philosophy, one that we also see present in Plato's Phaedo in a different context: there, he argues that all change is between opposites, with the result that the living shall die, the die shall come back to life, etc.

As for Aristotle:

In On Generation and Corruption, he explains that the four elements of the world around us, earth, water, air and fire, (as opposed to the fifth element, which belongs only in the realm beyond the moon, aether) are able to change into one another. For instance, example, fire can become air just by becoming colder, and earth can change into water when it changes from being dry into being humid.

The most obvious instance of such a transformation is the cycle of water as Aristotle explains it in the Meteorology: the heat of sun turns into vapour the water which is on the surface of Earth in seas, lakes or rivers.

This is cyclical because this vapour rises into the sky where it can condense into clouds and give birth to various meteorological phenomena such as rain, snow or hail, thus returning to the ground for a new cycle.

Different regions might be at different points in this cycle at different times, so there are no cosmos-wide catastrophes, which is a big difference between Aristotle and the Stoics.

Nevertheless, Aristotle does think that these changes can be huge from the point of view of any given human society. Indeed, part of this was built into ancient Greek cultures and myths, such as the myth of Deucalion, a man who survived a great flood and became an ancestor to the whole Hellenic people. Aristotle explains:

‘There happens at some fixed intervals of time, just as winter in the succession of each year’s seasons, a great winter and excess of rain recurring after a long cycle; and this excess of rain does not occur always in the same regions, but in the same way as what is called ‘‘Deucalion’s deluge’’; for the latter happened mainly in the Greek region, especially in the most ancient part of Greece, i.e., that around Dodona and the Achelous river ［in Western Greece］(Meteor., I, 14, 352a 29-35).

These climate cycles present large-scale difficulties to human civilizations. He is very sensitive to the way that humans are fragile and are at the mercy of the climate:

At the time of the Trojan war, the region of Argos could feed only few people because of its being marshy, whereas that of Mycenae was doing well and as a result was held in greater honour; but now it is the contrary because of what we said before: the latter has become sterile and completely dry, whereas in the other the land, which had been sterile because of stagnant water, has now become usable（Meteor., I, 14, 352a 9-14).

Now we can turn to the Stoics, who think that the whole cosmos will be cyclically destroyed and then regenerated. It isn't a complete destruction in the sense of erasing it from existence but rather, it goes through a fiery conflagration that burns it to a crisp and then it is reborn from the ashes.

At certain fated times the entire world is subject to conflagration, and then is reconstituted afresh (Aristocles, quoted in Eusebius, Evangelical Preparation 15.14.2)

Here is the argument of Zeno of Citium, who states that the all will be subject to conflagration: 'everything which burns and has something to burn will burn it completely: now the sun is a fire and will it not burn what it has?' (Alexander Lycopolis 19.2-4)

On the world's periodic destruction into fire at very long intervals, 'destruction' is not used in an unqualified sense by those who hold that the whole world is dissolved into fire, which they call the conflagration. They use the term 'destruction' in place of natural change.

Why? Well, there are a couple of moving parts here.

The first is that the Stoics believe that the world is alive. In Greek philosophy, living things are those things that move themselves, and living things do this in virtue of having a soul. Since all 'having a soul' means is 'being able to move yourself' or 'being alive', it is possible for a person to believe that I have a soul and that soul is corporeal and mortal. Philosophers who hold that the soul is immaterial and immortal (e.g., Plato) usually have to argue for this at great length. The Epicureans, for instance, argue that the soul is mortal and corporeal. But nobody denies that we have souls -- it'd be like denying that life exists.

The Stoics observe that the whole world moves -- after all, if you're in a geocentric universe, then you think that the whole cosmos is moving around the Earth. (And indeed, we know that whole galaxies rotate -- just not around the Earth!)

So, if the whole cosmos moves itself, then it must have a soul.

And moreover, the whole cosmos clearly has a body: it's all the corporeal stuff that we see around us.

Therefore, the cosmos has a body and a soul.

It doesn't follow that the soul is immaterial and immortal, though: the Stoics think that the humans have a soul that will die in the conflagration, for instnace. But it's material, too: depending on which Stoic you read, it's either made of breath or made of fire.

But the soul that animates and moves the whole cosmos is very fiery. And it's the fire that will ultimately burn up the whole cosmos.

But why do I bring this up? Well, the reason is that since the Stoics think that the whole cosmos has a body and a soul, then it is a living thing like we are.

Just as living things have a life-cycle that is witnessed in parents and then again in their off-spring, so too the universe has a life cycle that is repeated. There is a cycle of endless recurrence, beginning from a state in which all is fire, through the generation of the elements, to the creation of the world we are familiar with, and eventually back to the state of pure designing fire. This is meant to be like the cycle of any given living thing: we are born, we mature, and then we die. The cyclical rebirth of the cosmos is akin to the way that one generation dies -- and hands the baton to the next.

Some Stoics talk about how everything that happens will repeat exactly as it has already happened, which means that you, literally the exact same you will go through your entire life again, and again and again:

[The Stoics say that］ the primordial fire is like a kind of semen, containing the developments and causes of all things past, present and future, so that their combination and consecution is the inevitable and ineluctable fate, knowledge, truth and law of beings (Aristocles of Messene, De phil., fr. f3, 15-19)

The Stoics believe that after the conflagration everything will be recreated in the world numerically the same, so that even the individual with its particular characteristics will be born and exist the same in the next world as in the previous one, as Chrysippus says in his books On the World ( Alexander of Aphrodisias, In Anal. Pr., 180, 33-36:)

You will be reading this subreddit again. A long time from now...

The closest thing Greco-Roman mythology has to an end-time event is likely the conclusion to the “Ages of Man” arc of human history. These were described both in Greek mythology by Hesiod in “Works and Days” and in Roman mythology by Ovid in “Metamorphoses”. Hesiod described five ages while Ovid described four, but both are similar so I will walk through Hesiod’s to give you an idea (you can skip to the Iron Age for a short answer to your question):

Golden Age – first humans created by Kronos. Humans are almost god-like as they live in a perpetual spring, do not age and do not feel pain. But this age ended when Zeus defeated and killed Kronos.

Silver Age – When Zeus took over, he created the second age of Earthly beings. These humans were much more inferior to the gods when compared to the humans of the Golden Age but they still lived for a long time. Notably, this is when the four seasons were introduced so they had to work to save food. However, these humans refused to give sacrifices to the gods so Zeus destroyed them.

Bronze Age – These humans were bronze or red, in the sense that they were angry and violent. They loved war and eating meat. The end to this age is the famed flood myth, where the world was flooded, and all humans died except for Prometheus’ son Deucalion and his wife Pyrrha who was Pandora’s daughter.

Heroic Age – This age sits between the Bronze Age and the current age. This is the age where many of Homer’s stories took place including the Iliad and the Odyssey, as well as other myths such as the Argonauts and Oedipus. Many humans during this age were considered superhuman although not divine. The Heroic Age ended with the end of Mycenaean Greece during the Bronze Age collapse and in mythology it ended shortly after the Greeks returned from Troy.

--- Short answer to your question ---

Iron Age – This was the age Hesiod said he lived in and according to him it is the final and worst age. This is because humans suffer and strife exists between almost all humans including family. The Iron Age will decline as humans will no longer feel shame for wrong-doing, babies will be born with grey hair and all gods leave Earth. It will end when Zeus finally destroys this race although not much detail is given beyond that.

Tl;dr Humans begin as god-like beings but with each successive age up to the current one they decline with shorter lives and more evil. In this age humans will continue to decline until Zeus destroys the entire race.

Sources:

Hesiod., Tandy, D., & Neale, W. (1996). Hesiod’s Works and days . University of California Press

Fantham, E. (2004). Ovid’s Metamorphoses . Oxford University Press.

A point form description of the Five Ages of Man put together by Professor Phillip V. Stanley at SFSU: http://online.sfsu.edu/pstanley/clas360.htm/myth5b.htm