Short answer: no.

Long answer: numerous elements of the setting are certainly supposed either to be real or else have verisimilitude. Book 4 is set in Sparta, which is a real place; the Ionian Islands where Odysseus lives are real islands (though identifying which one is supposed to be Odysseus' Ithakē continues to be a debated topic); elements of Ithakan society are either real (the assembly and its role as a decision-making body, the process of arbitration) or intended to have verisimilitude (it's doubtful whether hospitality, or the position of basileus, ever worked the way they do in the Odyssey, but it seems that Homer's audience at least believed that's how things had once worked).

Some specific items:

• Ithakē is a real place, and so are the neighbouring islands of Samē (the Homeric name for Kephallēniē, or modern Kefaloniá) and Zakynthos. Doulichion is probably a real place too, though it cannot be identified securely. The Odyssean description of Ithakē does not fit the modern island of Itháki, so it has often been wondered whether it was supposed to be a different island, or perhaps a peninsula of Kefaloniá (this is Robert Bittlestone's theory); or, on the other hand, whether the poet simply didn't know the area and so got the description wrong (and this theory is much more parsimonious).

• Pylos and Sparta are also real (though the identification of Pylos is also problematic).

• However, pretty much all other places are fictional, or at least heavily fictionalised. People in antiquity did like to equate them with real places. Typically, Scheria (the island of the Phaiakes) was usually thought to be Kérkyra; the lands of the Kyklopes and Laistrygonians were thought to be in Sicily; Aiolos' island was thought to have been Lipari, in the Isole Eoli just north of Sicily; Kirke's island, the consultation of the dead in Odyssey 11, and the Sirens were thought to have been in central Italy; etc., etc. The rationale for these identifications was that before Greeks colonised these lands (beginning in the 8th century BCE), they had supposedly been wildernesses filled with weird monsters and supernatural entities. It's possible that they were all intended to be set at these real locations all along, but even if so, the descriptions in the Odyssey are very poor matches for the real places, and they are even more fictionalised than Ithakē itself is. To all intents and purposes they're not real.

Oddly, native Italians loved having these Greek stories set in Italy, and the Etruscans took to Greek mythology like a duck to water. It's a little hard to see exactly why: it seems that it provided a useful basis for an awful lot of Greek-Etruscan trade. That didn't really happen anywhere else, though. But love it they did, and from the seventh century onwards Etruscan art is packed with legendary figures from the Trojan War. The Greek characters acquired etruscanised names: so for example Odysseus became Uthusthe, Uthuze, or Utuse; Kirke became Cerca; Polyxenē became Phulsphna; Diomedes became Ziumithe and supposedly founded a few Samnite cities in inland Campania. The emergence of Rome in the sixth and fifth centuries continued this tradition and also gave a fairly high level of prestige to Greek mythological figures, like Castor and Pollux, Hercules, Ulysses/Odysseus, and Aeneas; in the sixth century Rome founded a colony on the supposed site of Kirkē's island, and called it Circeii.

• In terms of culture and politics, the Odyssey takes care to have verisimilitude in its portrayal of a distant past, but it does so by means of false archaism rather than by recording anything that was actually real in the past. There's no evidence that hospitality, or xeiniē, as practised in the Odyssey was ever a real thing, for example: instead, the narrative takes a process that would have been familiar to people of the time as a religious rite, called theoxenia, and substituted mortal visitors into the role played by a god in the religious rite. I've written a longer post about hospitality here; and one on false archaism here.

• Developments in the not-too-distant past:

• As mentioned above, Robert Bittlestone has the theory that the land called "Ithakē" in Homer is actually what is now a peninsula of Kefaloniá, Palíki. I don't have the geological chops to evaluate this theory, though I will note that it requires that there was once a channel between Palíki and the rest of Kefaloniá that was about 8 km long and never more than 100 m across. Hm.

• In 2008 an article appeared claiming that a very figurative speech in Odyssey 20, referring to "darkness blotting out the sun", referred to a historical solar eclipse that took place in 1178 BCE. One of the authors was quoted in the press at the time as saying that this proved "everything else described in the Odyssey happens exactly as is described"... though that might be a misquotation.) This theory was never given any credit within the field, and an article appeared a couple of years ago claiming to debunk it. (For the record, four lines after the relevant passage, another speaker contrasts the darkness inside the hall with the sunshine outdoors.)

The Odyssey is very much a creation of its time and tailored to circumstances of its time. The culture it describes is that of the 8th-7th centuries BCE, modified by false archaisms to create the impression of antiquity. Odysseus' wanderings may or may not have been designed with a setting in contemporary Italy and Sicily in mind; either way, their source material predominantly comes from the legend of the Argonauts (as shown in Georg Danek's 1998 book Epos und Zitat: Studien zu den Quellen der Odyssee). Odysseus and his family were popular legendary figures both in Greece and Italy, but there are definite historical reasons for this, and a legend's popularity is never an indicator of its truthfulness.