I answered a similar question a few months ago (which was actually compiled from a couple other answers made to similar questions). I'm copying it here, with a couple minor edits.

There really was a Bronze Age conflict which probably inspired Homer's epic.

Did any of the people in the story exist? No evidence for that exists. Did events happen as described in the story? Probably not, it was probably very embellished (like historical fiction movies/books). But we know from Hittite archives that there was a conflict between the city of Wilusa (the Greek name for Troy is Ilion, which is a cognate/loan word from the Hittite name for the city, Wilusa) and a people group called the Ahhiyawa (often identified as the Achaeans). The specific text refers to something called "the Wilusa episode" which involved hostility on the part of the Ahhiyawa toward Wilusa, which took place around the 1300s-1200s BC. All ancient textual indications and current scholarly consensus is that the Ahhiyawa are from the area of Greece/Mycenae.

The identification of Wilusa as Troy is also pretty sound, historically and archaeologically. All textual references that locate the city geographically put it in the area that the mound of Hisarlik sits, and there are mentions of such things as a water tunnel in Hittite texts, which have been found in excavation at Troy. So it's pretty well-accepted that Troy=Wilusa=Hisarlik.

So we know that there was a city called Troy, and a people called Achaeans, and that the Achaeans attacked the city of Troy. In addition, one of the kings of Wilusa mentioned in Hittite texts was Alaksandu, which is the Hittite version of Trojan prince Paris' birth name (Alexandros). However, it is very unlikely that it refers to the same person, but we know there were people with similar names to Homer's epic who did exist.

However, Homer wrote his epic centuries after it took place, between 1102 BC (Pseudo-Herodotus' date) and around 850 BC (Herodotus' estimation). We have no indication that it was intended to be an accurate retelling of events, or anything more than a fun imagining based around something that actually happened (like Inglourious Basterds and WWII, or "Cold Mountain" and the Civil War). All we know is that yes, there was a war, and the Achaeans attacked Troy, and that's about it.

For further reading, I suggest

Sagona, A G, and Paul E. Zimansky. Ancient Turkey. London: Routledge, 2009

Easton, D.F.; Hawkins, J.D.; Sherratt, A.G.; Sherratt, E.S. (2002). "Troy in Recent Perspective". Anatolian Studies (52): 75–109

Allen, Susan (July 1995). "'Finding the Walls of Troy': Frank Calvert, Excavator". American Journal of Archaeology 99 (3): 379–407

Wood, Michael (1985). In Search of the Trojan War. BBC Books; First Thus edition

Just a heads up:

Did it actually happen?

Is fine!

Who would win, Achilles and 20 of his Myrmidons, or Hector and 30 of his elite warriors?

Really should be in /r/HistoryWhatIf or perhaps more appropriately, /r/whowouldwin.

The short answer is that no one really knows for sure. The longer and better answer is that the best estimate is that there really was a conflict (perhaps a trade dispute) between those we now term as "Greeks," but who at the time were part of the Mycenean culture, and a likely-Hittite people based in and around the archaeological site of Hisarlik, which we refer to as Troy.

This conflict probably took place in the Bronze Age, sometime within 100 years either side of 1200BC. This is based on several types of evidence ranging from various cultures. First, from the Iliad itself, is the so-called Catalogue of Ships, which names all the city-states/kingdoms that sent ships and contingents of warriors to the conflict. Some of the cities mentioned therein were in existence during the Bronze Age, but were not rebuilt following the collapse of those cities and entry to the period of the Greek dark age. (However, it should be noted that some other scholars believe that the Catalogue is more representative of the political situation in Homer's time, around 800BC).

Second, Hittite evidence references place names that appear to be Hittite translations of the places we know from the Iliad. Specifically, Wilusa (for Wilios, or Ilios) and Taruisa (For Troias, or Troy). Additionally, the Hittite sources make reference to a group of people known as Ahhiyawa (or Achaeans), and to specific people such as a Wilusan king named Aleksandru (or Alexander, which is the alternate way that Paris is referred to in the Iliad). This all suggests that the Hittites did have some contact with people from Greece, although it is certainly not definitive.

Third, Mycenean Linear B tablets from the Bronze Age reference someone named Achilles. Although this person is not nobility, the fact that the name itself exists suggests a time period for when the event might have taken place (or at least for generation of the oral form of the story about the event).

Finally, excavation of the Hisarlik site, first done by Heinrich Schliemann in the late 1800s, has found numerous layers of settlement at the location. Assuming Hisarlik is in fact Troy, the most likely layer to be involved in the Trojan War is Troy VIIa: it is of an equal time period as the Hittite sources referring to a place as Wilusa, and it appears to have been destroyed in a war, with archaeological evidence in the form of skeletons with injuries, fire, and related indicia of violent conflict.

With the caveat again that we can't know for sure, all of this evidence suggests that some type of conflict did happen, and that the general timeframe for such a conflict is around 1200BC.

Sources:

The War that Killed Achilles - Caroline Alexander

Travelling Heroes in the Epic Age of Homer - Robin Lane Fox

Others which I can't remember because I'm away from my bookshelf.