Within hours of taking the throne, in August AD117, the emperor Hadrian made one major strategic decision. He issued the order to withdraw the Roman troops from Iraq (or Mesopotamia, as he would have called it). His succession had been a messy one, in the usual Roman way. Despite a well-earned reputation for effective administration in most areas, the Romans never really sorted out the transfer of imperial power. Hadrian's leadership bid was more reminiscent of what goes on in the Labour party than in the House of Windsor. It involved a good deal of manipulation, double-dealing, back-stabbing (in Rome this was real, not metaphorical) and perfect timing. A couple of rivals had made their bid too soon, leaving Hadrian as the only plausible candidate to be adopted by his elderly predecessor Trajan, just a few days before he died.

Hadrian was instantly faced with a problem in the Gulf. Trajan had sponsored ambitious expeditions in the east - determined to get control of the rogue states threatening Roman interests there, and in his wider dreams to follow in the conquering footsteps of Alexander the Great. He had reached the enemy capital at Ctesiphon, just south of modern Baghdad, where he made his own premature declaration of "Mission Accomplished" (in Latin, "Parthia capta" - a phrase blazoned across the commemorative coinage). He had then moved on to Basra, where he planted the Roman flag, and sensibly decided that he was too old to take the Alexander trail to India.

This whole enterprise was already going horribly wrong before Trajan's death in 117. He had tried the trick of restoring some form of local control in Ctesiphon, in the shape of a puppet king (another series of coins vainly celebrated the restoration of constitutional government, much as we have celebrated the restoration of Iraqi "democracy"). But the rival factions and insurgencies undermined all attempts to bring peace and order. Hadrian saw the impossibility of the task and straightaway pulled the troops out, leaving the various local warlords to fight it out themselves.

He diverted the legions to more winnable campaigns elsewhere. There was unrest, as usual, in the Balkans. And in the near east he had to finish stamping out a Jewish revolt which, according to some wild and fearful Roman estimates, had cost half a million Greek and Roman lives. Fifteen years later, prompted among other things by a recent ban on circumcision, the Jews rebelled again under Shimon bar Kokhba. Charismatic or charlatan, depending on your point of view (the predictably hostile Saint Jerome later claimed that he "fanned a lighted straw in his mouth so that he appeared to be breathing out flames"), he commanded a force that was at first a match for the Romans. In the end, Hadrian's forces had to resort to the most ruthless form of ethnic cleansing, constructive starvation and mass slaughter of the enemy that went far beyond the casualties inflicted by the Jews. In Rome, and among generations of antisemitic ideologues up to the 20th century, the victory was hailed a triumph over religious fanaticism and political insurrection.

The new exhibition at the British Museum, Hadrian: Empire and Conflict, features evocative objects from both sides of this Jewish war. There are simple everyday items recovered from a Jewish hideout: some house keys, a leather sandal, a straw basket almost perfectly preserved in the dry heat, a wooden plate and a mirror - evidence of the presence of women, according to the exhibition catalogue (as if men did not use mirrors). But with or without the women, these are all bitter reminders of the daily life that somehow managed to continue, even in hiding and in the middle of what was effectively genocide. From the other side, there is a magnificent bronze statue of the emperor himself, which once stood in a legionary camp near the River Jordan. The distinctive head of Hadrian (bearded, with soft curling hair and a giveaway kink in his ear lobe) sits on top of an elaborately decorated breast-plate, on which six nude warriors do battle. It is a striking combination, even if - here as elsewhere - the catalogue raises doubts about whether the head and body of this statue originally belonged together.

Far away from Judaea, on the other side of the Roman world, Hadrian's military operations in Britain were less bloody. Apart from the low-level guerrilla warfare endemic in most Roman provinces, he had his troops occupied in building the famous wall running across the north of the province. This was a project inaugurated when Hadrian himself visited in 122, one of the few Roman emperors ever to set foot in the empire's unappealing northern outpost. It is now far from certain what this wall was for. The obvious explanation is that it was built to prevent hordes of nasty woad-painted natives from invading the nice civilised Roman province, with its baths, libraries and togas. But - leaving aside the rosy vision of life in Britannia that this implies (baths, libraries and togas for whom exactly?) - this overlooks one crucial fact. The impressive masonry structure, which provides the iconic photo-shot of the wall, makes up only part of its length. For one-third of its 70 miles the "wall" was just a turf bank, which would hardly have kept out a party of determined children, never mind a gang of barbarian terrorists.

There are all kinds of alternative suggestion. Was it, for example, not much more than a fortified roadway across the province? Or was it more of a boast than a border - an aggressive, but essentially symbolic, Roman blot on the native landscape? Most likely it was for the control rather than the exclusion of people. The aim was to channel regular movement into certain standard crossing points (even the turf bank would have been inconvenient to cross with a loaded cart), to police the migration of people both ways, and possibly also to tax the goods that came and went. On the spectrum of modern walls, that would make it closer to the Mexican border fence than to the Berlin wall.

If all this seems rather familiar, that is partly because there really are significant overlaps between the Hadrianic empire and our own experience of military conflict and geopolitics. We are still fighting in many of the same areas of the world and encountering many of the same problems. We are still claiming victory long before we have won the war - or indeed, in the Iraqi case, instead of winning the war. We still turn to masonry (plus, in the modern world, barbed wire) to separate one arbitrary nation from another and to police arbitrary boundaries. It is not going too far to suggest that there are political lessons we can still learn from the failure, or success, of Roman enterprises in the Gulf and elsewhere.

But there is a more complicated and interesting story here, too. For Hadrian himself has long seemed a familiar figure in many other respects. He is not exactly "one of us", perhaps, but he is at least one of those rare characters from the Roman world to whom even now we can feel quite close. In contrast to the sheer madness of Nero or Caligula, or to the disconcerting and implausible probity of the first emperor Augustus, Hadrian is the kind of political leader whose behaviour seems distinctly recognisable, whose ambitions and conflicts we can almost share.

That feeling of familiarity has been boosted by Marguerite Yourcenar's fictional, pseudo-autobiography of the emperor, Memoirs of Hadrian. Published in 1951, and once hugely popular (it now seems to me rambling and frankly unreadable), it took the modern reader inside Hadrian's psyche - presenting the emperor as a troubled and intimate friend, in much the same way as Robert Graves made the emperor Claudius a rather jolly great-uncle. But Yourcenar's fictional construction is not the only reason for Hadrian's apparent modernity. There are all kinds of ways in which Hadrian's life and interests seem to match up to our own expectations of monarchs and world leaders, and to modern interests and passions. He was the sponsor of Mitterand-style grands projets, a great traveller to the outposts of his dominion (including that trip to Britain), as well as an enthusiastic collector of art. And to cap it all, he had an intriguing, and ultimately tragic, sex life.

The British Museum exhibition makes a good deal of his building work and his art collecting. Unsurprisingly, perhaps, given that the museum itself is the descendant and direct beneficiary of Hadrian's passion for architectural design and classical sculpture. His most famous building in Rome was the great Pantheon. One of the few ancient Roman buildings to remain standing to its full height, and even now in active use as a church, it is crowned with what is still the largest dome ever built with unreinforced concrete. This has been the inspiration behind almost every great dome built since, from St Sophia in Istanbul (a grand projet of one of Hadrian's eastern successors, the emperor Justinian) to the dome of the museum's own round reading room. By a nice symmetry, it is here that the Hadrian exhibition has been displayed - placing the emperor, so to speak, in his own dome.

It is also the case that a substantial part of the museum's collection of Roman sculpture came from what is known as Hadrian's "villa" at Tivoli, some 20 miles outside Rome. This was in fact a vast, sprawling palace and pleasure gardens built by the emperor, occupying the space of a large Roman town (it is at least twice as big as Pompeii). Here Hadrian created an extraordinary microcosm of his own empire, replicating in miniature all kinds of famous landmarks and artistic masterpieces from across the Roman world. The lovely long pool that is a highlight of the site for modern visitors seems to have been a version of a celebrated Egyptian waterway, the Canopus canal. In another part of the palace, he not only had a copy of one of the most renowned Greek statues, the fourth-century BC nude Aphrodite from the town of Cnidus, by Praxiteles (reputed to be the first Greek female nude ever), but he displayed it within a replica of the very temple in which she was kept in Cnidus. The "villa" offered, in Roman terms, a vision of universal culture, not wholly different from British Museum director Neil MacGregor's idea of the "universal museum".

It also housed an enormous quantity of sculpture. And Tivoli, unlike many of the crucial areas of the city of Rome itself, was not built over in the centuries that followed the fall of the empire. From the 17th century on, the site was an easy gold mine for archaeologists, collectors and art dealers in search of antiquities to draw, to sell or take home (you can still see on one of the villa walls Piranesi's signature, scrawled in red pencil during a drawing expedition in 1741). There was plenty of stuff to go round, and a number of major European sculpture collections were formed around a nucleus of material that had been excavated at Tivoli. Among them was the collection of Sir Charles Townley, most of which was bought by the British government in 1805 and became the basis of the British Museum's Greek and Roman collection. Several of Townley's pieces are on show in the new exhibition, including a Hadrianic relief of a boy with a horse, obviously inspired by the Parthenon frieze - which was in Hadrian's day still in its original place on the Parthenon. This is a wonderful vignette of the complex history of collecting, and its surprising overlaps and intersections. Not only do we find the collection of Hadrian becoming part of the collection of Townley, and then of the British Museum. But whatever your view on the repatriation of the Elgin marbles, it is hard not to be struck by the marvellous irony of Hadrian's copy of the Parthenon frieze ending up in the same museum as much of the original.

Another major theme of the new show is Hadrian's relationship with Antinous, a boy who came from Bithynia, in modern Turkey. We know no details whatsoever of what went on between the two, but the usual story - misogynist as so many such stories are - contrasts the emperor's passion for this beautiful lad with the loveless, childless marriage to his bad-tempered and scheming wife, Sabina. What is certain is that Antinous died young, drowned in AD130 in the River Nile (murder, esoteric sacrifice, suicide and tragic accident have all been suspected), and that following his death Hadrian devoted enormous energies to his commemoration. He had him made into a god. He founded and named a city after him, Antinoopolis, on the banks of the Nile where the boy had drowned. At Tivoli, near one of the main entrance-ways to the palace, he greeted visitors with an elaborate cenotaph for Antinous, in distinctive Egyptian style - complete, it seems, with palm trees.

He also flooded the Roman world with his statues. About a hundred portraits of Antinous are known, more than we have for any other individual Roman, apart from the first emperor Augustus and Hadrian himself. These come in all shapes, sizes and styles, from colossal images in the guise of an Egyptian god to precious miniatures in silver. But the standard, instantly recognisable type is of a languorous young man, pouting, heavy-lipped and sultry - an image that has come to be almost a shorthand for "sex in stone". It is perhaps no surprise that JJ Winckelmann, the 18th-century art historian, archaeologist and homosexual, steamed over one particular sculpture of the boy in a private collection in Rome. In fact, the most famous portrait of Winckelmann shows him studying an engraving of that very statue. But even now the sight of Antinous can work its magic. One of the portrait heads in the British Museum exhibition is a vast sculpture from the Louvre, known as the "Mondragone Antinous", after the place in Italy where it was first put on show in the early 18th century. Although a few recent critics have gone against the grain and deemed it a faintly repulsive, pouting monstrosity, others have made no secret of their admiration. When it was unpacked from its crate in Leeds a few years ago, where it was due to star in an exhibition devoted to Antinous at the Henry Moore Institute, it bore on its cheek the clear traces of a bright red lipstick kiss.

Traveller, patron, grief-stricken lover, art collector, clear-thinking military strategist. How do we explain why Hadrian seems so approachably modern? Why does he seem so much easier to understand than Nero or Augustus? As so often with characters from the ancient world, the answer lies more in the kind of evidence we have for his life than in the kind of person he really was. The modern Hadrian is the product of two things: on the one hand, a series of vivid and evocative images and material remains (from portrait heads and stunning building schemes to our own dilapidated wall); on the other, the glaring lack of any detailed, still less reliable, account from the ancient world of what happened in his reign, or of what kind of man he was, or what motivated him.

The only fully surviving ancient biography is a short (20 pages or so) life - one of a series of colourful but flagrantly unreliable biographies of Roman emperors and princes written by person or persons unknown, sometime in the fourth or fifth centuries AD. This includes one or two nice anecdotes, which may or may not reflect an authentic tradition about Hadrian. My own particular favourite features his visits to the public baths. The story goes that on one occasion Hadrian spotted a veteran soldier rubbing his back against the marble wall. When he inquired why he did this, the old man replied that he could not afford a slave. So Hadrian presented him with some slaves, and with the money for their upkeep. On his next visit, there was a whole crowd of old men rubbing their backs against the wall. Far from repeating his gift, he suggested that they take it in turns to rub each other down. There were a number of morals here. Hadrian was a man of the people, not above mixing with the plebs in the public baths. He had his eyes open for his subjects' genuine distress and personally intervened to help. But you couldn't take him for a ride.

Sadly, very little of the life is up to this quality. Most of it is a garbled confection, weaving together without much regard for chronology allegations of conspiracies, accounts of palace intrigue, and vendettas on Hadrian's part - plus an assortment of curious facts and personal titbits (his beard, it is claimed, was worn to cover up his bad skin). To fill the gaps, to make a coherent story out of the extraordinary material remains of his reign, to explain what drove the man, modern writers have been forced back on to their prejudices and familiarising assumptions about Roman imperial power and personalities. So, for example, where - thanks to the surviving ancient literary accounts - it has been impossible to see Nero as anything other than a rapacious megalomaniac, Hadrian has morphed conveniently into cultured art collector and amateur architect. Where Nero's relationships with men have to be seen as part of the corruption of his reign, Hadrian has been turned into a troubled gay. Hadrian seems familiar to us - for we have made him so.

The British Museum exhibition presents Hadrian as an appropriate successor to the first emperor of China and his terracotta army, both key figures in the foundation and development of early imperial societies. Maybe so. But an even better reason to visit this stunning show is to see how the myth of a Roman emperor has been created - and continues to be created - out of our own imagination and the dazzling but sometimes puzzling array of statues, silver plates and lost keys of slaughtered Jewish freedom-fighters.

· Hadrian: Empire and Conflict is at the British Museum, London WC1, from July 24 to October 26. Box office: 020 7323 8181, boxoffice@britishmuseum.org