Troy is a real place. The excavated city of tumbled stone at Hisarlık in western Turkey, near the mouth of the Dardanelles, is generally accepted as the site of its remains. But perhaps Troy is now more a zone of the imagination, rebuilt and repeopled every time someone makes it afresh in their mind – through the act of reading or looking, writing or making.

It all started with the Homeric poems: The Iliad, about the Trojan war; and The Odyssey, about Greek fighter Odysseus’s troubled, circuitous return home after victory. Dionysius of Halicarnassus, the first century BC scholar, called Homer the source from which every sea, every fountain, every river flows. As if to prove his point, he was quoting Homer when he wrote that (specifically, the ancient Greek poet’s description of the ocean, the encircling girdle of waters that surrounds the known world, in book 21 of The Iliad).

The poems – drawing on many years of orally transmitted song and written down somewhere between the late 8th and late 7th centuries BC – loom inescapably large in later classical literature. Sappho remade Troy in her heartstopping love poetry, allowing the sexy, semi-divine, magnetic figure of Helen, over whom the war was fought, to remind her of her own alluring lover. The Greek tragedians rebuilt Troy and the places connected with it in such plays as Hecuba and Iphigenia in Aulis and Agamemnon. Virgil recreated it magnificently as a Roman national epic.

People are still remaking it, including in recent years the writers Seamus Heaney, Pat Barker and Alice Oswald. Artists have been similarly drawn to the tale. In Troy: Myth and Reality, a new show at the British Museum, African American artist Romare Bearden reimagines, in a collage from the 1970s, Odysseus as travelling through African and Caribbean realms. In a photographic work from 2007, Eleanor Antin turns The Judgement of Paris, one of the events that led to the war, into a camp beauty contest in which Athena, goddess of war and winning, poses like Lara Croft, while Hera, protector of the family, vacantly pushes a vacuum cleaner in 1950s frock and pinny.

Immense and phallic … The Vengeance of Achilles, 1962, by Cy Twombly. Photograph: Kunsthaus Zürich

The show begins with a striking variety of objects, only two of them from the Dardanelles. Straight ahead is Cy Twombly’s immense 1962 drawing, The Vengeance of Achilles, a phallic, raging, scarlet-soaked triangle that makes me think of Christopher Logue’s Iliad-inspired poem, All Day Permanent Red. To the right is a trio of Anthony Caro sculptures from the 1990s called The Death of Hector, King Priam and The Skaian Gate. They are poised, monumental tableaux in wood, ceramic and steel that fix moments from the Trojan story in solemn stillness.

In front of the Twombly are three ancient objects. One is a black-figure amphora, or vessel, made in Attica in Greece by the potter and painter Exekias in around 530BC. It depicts the Greek fighter Achilles, high-helmeted and staring-eyed, thrusting a spear into the pale, undefended throat of the Amazon queen-warrior, Penthesilea. The other two are simple pots from Hisarlık, excavated by Heinrich Schliemann in the 1870s. The pots are blackened, not by the burning walls of Troy, but by the burning walls of Berlin, where they were damaged by allied bombs in the second world war.

Troy is both fugitive and real, a fiction and a truth, a place of the deep past and an endless parade of future and possible cities, all magnificent and teeming, all broken and burned. None of the objects in the exhibition can be ascribed to a real Trojan war, because Homer’s Iliad is not a historical document – though it may be, on some level, a scrambled, dreamlike, poetic memory of raids or sieges, of conflicts between Greeks and Trojans that took place in the previous millennium.

We are all Trojans. For centuries, Homer’s Iliad was lost to western Europe. In the medieval era, until Petrarch and the seeds of the Renaissance, it was Latin that was read. So it was Roman stories of Troy that circulated: most importantly, The Aeneid. This poem, which Virgil was working on at the time of his death in 19BC, is a kind of sequel to The Iliad and The Odyssey, and a Roman rival to them. It tells the tale of Aeneas, a Trojan prince whose destiny is to escape the defeat of Troy, to leave the flames, rape and death behind – and find Italy, where he will establish a new dynasty that will end up founding Rome, the second Troy. In medieval Europe, the idea of one’s nation having been established by Trojan exiles, a la Rome, was irresistible. Every nation had to have a Trojan founder.

On the hoof … a scene from Wolfgang Petersen’s 2004 film Troy. Photograph: Allstar/Cinetext/Warner Bros

Britain’s is Brutus – the clue is in the name – who is the supposed great-grandson of Aeneas. Unmentioned in any antique source, he pops up in Geoffrey of Monmouth’s 12th-century The History of the Kings of Britain (more myth than history). When Brutus arrives, he changes the name of Albion to reflect his own. The city he founds is Troynovant, or the new Troy, which later becomes London. “For noble Britons sprong from Troians bold,” wrote Edmund Spenser in The Faerie Queene. “And Troynount was built of old Troyes ashes bold.” In one genealogy, the Lyte Pedigree of 1605, James I’s ancestry was traced directly back to Brutus.

An exhibition attempting the gargantuan task of recovering the archaeology of Troy, its framing in antiquity and its immense afterlife, can only gesture towards completeness. The vast subject of Troy and the Renaissance, for example, is represented by very few objects, including maiolica plate, a Rubens picture and a single tabletop version of the endlessly copied Hellenistic sculpture of the Trojan priest Laocoön that was excavated in 1506 in a vineyard in Rome, with Michelangelo and a couple of members of the Sangallo family present. (Like the Bloomsberries, Renaissance artists always seem to have been popping in and out of each other’s houses.)

To me, the most arresting objects in this show (the “BP exhibition” as it is billed, ethically untenable given the sponsor is one of the world’s great polluters) are those from the Greco-Roman world. From the museum’s own collection is a magnificent dinos, or wine-mixing bowl, from about 580BC, signed by Sophilos and resplendent on its sphinx-decorated stand. It narrates the Trojan backstories – including the marriage of Peleus and Thetis, parents of Achilles, and The Judgement of Paris. Berlin’s Antikensammlung has lent a marvellous fifth-century BC red-figure skyphos, or cup, that shows Odysseus shooting his bow at the suitors who are hounding his wife, Penelope. The female slaves stand by as the feasting suitors collapse on their couch, impaled by arrows. One shields himself with an elegant stool.

Brutal … Achilles kills the Amazon queen Penthesilea, depicted on an amphora from around 530BC.

A pair of precious silver Roman cups from Denmark’s Nationalmuseet – unusual survivals, found in an iron age chieftain’s grave – are decorated with Trojan war scenes. On one, Priam begs Achilles for the return of his son Hector’s body; on the other, Odysseus is negotiating with Philoctetes, the Greek bowman, left behind on Lemnos owing to his hideous-smelling thigh wound – but his bow is now needed, the prophecies say, if Troy is to be taken.

There is a fresco panel from Pompeii, a subtle and painterly vision of Trojans fatally pulling the wooden horse into their city. There is a drop-dead gorgeous piece of Etruscan wall painting, from a sixth-century BC tomb, showing a particularly sexy Judgement of Paris, with Aphrodite showing off her thighs and all three goddesses sporting wonderful pointed slippers.

But maybe most touching of all – and at the same time, perhaps the least physically impressive artefact in the exhibition, given it’s cracked, broken and brown – is the object known as Nestor’s cup. It was found on Ischia, in the Bay of Naples, once the Greek colony of Pithecusae, and it usually lives in the little museum there. But it was made in the eastern Mediterranean, in what is now Anatolia. It is tremendously old – circa 715BC – and the writing on it is one of the earliest surviving examples of the Greek alphabet.

That script is three lines of verse, two in dactylic hexameter, the metre of epic verse. The lines translate as: “I am the cup of Nestor, good to drink from; whoever drinks from this cup, immediately desire of fair-garlanded Aphrodite will strike him.” Nestor is a character in The Iliad, a veteran Greek fighter, loquacious storyteller and strategist. The cup that is “speaking” here is invoking the drinker’s knowledge of and pleasure in the stories of Troy, joking and riffing on the tales of the old heroes. It seems that as long as Greece had writing, Troy was written about. And so it has been ever since, for almost 3,000 years.

• Troy: Myth and Reality is at the British Museum, London, 21 November to 8 March.