My partner, Greg Doran, and I have come to Turkey to celebrate our 20th anniversary. During those years, we’ve learnt that we favour two different kinds of holiday – Greg wants culture, I want sun and sea; we’re hoping Turkey will provide both.

We start at a terrific hotel in Istanbul, the Ajia, on the Asian side of the Bosphorus. Its main balcony, which belongs to our huge airy room, hangs right over the water, and there’s a particular exoticism to the traffic of great freighters and tankers passing by, travelling routes that their kind have for centuries, north to the Black Sea or south to the Aegean.

The hotel ferry takes us over to the European side, for a tour of Istanbul’s historic sites – Greg showing more appetite for this than I do. I find the Blue Mosque disappointing, expecting it to be, well, bluer and more mysterious. In pouring rain and with hordes of visitors, the Topkapi Palace doesn’t do much for me either. My favourite place is Haghia Sophia, a great hybrid beast of a building. First built as a Byzantine church by the Emperor Justinian in AD537, later converted into an Ottoman mosque, it now functions as neither. With no god in residence, its spiritual power becomes all the stronger for me, held somewhere in its vast, dusty, shadowy space, with golden mosaics making the dome shine in the gloom.

We travel on to the Oliviera Resort on Kalem, a privately owned island off the Aegean coast at Dikili. Ah, this is more my kind of holiday. Kalem is pale and stony like Greek islands, but handsomely planted with gardens and surrounded by a sea view from my dreams: a film of turquoise water over a patchwork of white sand and black rocks.

The young owner, Ali Dartar, takes us for a ride in his motorboat, pointing out the local sights: the neighbouring island, which is on sale for a mere £12m; the collapsed ruin of an early Christian monastery; and a thermal spring on the mainland coast. Here we dip in a bubbling, hot, sulphurous pool. Used since Roman times, it’s said to have healing powers, and, indeed, the pain from a recent calf injury vanishes.

Just as I’m happily settling into a sybaritic life – lazy swims, boozy lunches and long siestas – Greg announces that Kalem is well situated to visit several ancient sites, and we’re off on his excursions again. But these reveal breathtaking things.

On a strange, moody day, now spitting with rain, now glinting with sunshine, we visit Troy. There’s not much to see – just the meagre remains of the nine ages or layers of the city, all overshadowed by one great mythic layer: Homer’s epic tale The Iliad. “There’s where Achilles rode his chariot,” says our guide, “dragging the naked corpse of Hector.” As we speak of these imaginary wars, it is moving to realise that Troy’s broken old walls, splashed with the red of wild poppies, overlook a place of real warfare – Gallipoli, where more than 100,000 lives were lost in 1915. When the campaign was over, Ataturk, the victorious commander (and father of modern Turkey), said, “We have taken Hector’s revenge.”

Perched high on a hilltop, the citadel of Pergamon is said to have the steepest amphitheatre in the world. The 10,000-seat auditorium seems to hang in midair, with a vertiginous view of the town of Bergama on the plains below. Whether as audience or performer, how could your imagination not soar in this place?

The amphitheatre at the city of Ephesus is even bigger, seating up to 25,000. We are delighted to find what would have been the stage door. Did this dark passage of low pillars once house a wisecracking stage doorkeeper and a noticeboard with information about rehearsal calls and local restaurants? Some parts of Ephesus are remarkably preserved, almost as good as Pompeii, granting vivid glimpses of the past. I was intrigued by the unisex public toilets – your long robes provided modesty – where musicians would play (not, I guess, the best gig in town), and by the secret tunnel that linked the library to the brothel, allowing you to pretend to be in one while frequenting the other. Nourishment for brain or body – it’s like the theme of our holiday. WE’RE BOTH equally thrilled by the next stage: a private cruise along Turkey’s Mediterranean coast aboard a gleaming vessel called Serenity 70, a cross between a modern motorised yacht and a traditional sail-driven gulet. It could sleep eight guests, but it’s entirely ours for the next week. There’s a crew of three: Adam the captain, Mehmet the chef, and Ozgur the deckhand.

On the first morning, crossing the Gulf of Fethiye, we see dolphins. At first they seem to lay on a little display, leaping close to the yacht, then they suddenly swim right under the prow where we stand, rolling upside down in the translucent water, changing what was dark, solid and familiar – their arched spines and dorsal fins – into something pale, fluid and magical. “That’s a sign of luck,” Adam calls from the wheel. “I’ve never seen them here before.” I think: “Yeah, sure, bet you say that to all the tourists.” But later I believe it was true, for we don’t see dolphins again, and it certainly was a lucky thing to book this cruise.

The daily routine consists of sailing for a few hours in the morning before Adam anchors in some beautiful, unspoilt bay, lowers a flight of steps into the gorgeous green-blue water, and we bathe, calling to one another, as we did yesterday: “This is definitely the best swim of my life!” Then we lunch on another of Mehmet’s huge and delicious meals, which often include freshly caught fish, prawns and lobster. We’ve brought along our own stock of French wines, having discovered that Turkish wine is bizarrely expensive (an ordinary chardonnay costs £30). In the afternoon, we sail on, then moor in another bay for the night, where we swim again, and feast again, and ask again whether such luxury is not indecent.

OUR LAST hotel was Mehmet Ali Aga Konagi, a converted Ottoman mansion just outside Datca, where the Mediterranean and Aegean meet. Here we had one of the most remarkable hotel suites in the world. Alongside the bedroom is a reception chamber, a large open space, its walls lined with long, white divans, these filled with rushes and covered in embroidered cotton. The room is decorated with exquisite frescoes and delicately carved woodwork – it’s like being inside a jewel box – and the atmosphere is totally tranquil, with windows all around, their sunlit lace curtains lifting in scented draughts of air – jasmine and oleander – giving glimpses of the gardens and fountains beyond.

Greg immediately ensconced himself in one corner, among a heap of silk cushions and books, looking like some wise pasha of old, while I settled on the wide, wooden terrace outside, with a table for writing and sketching. And this is how we spent the week: waking to the sounds of the neighbouring village – cocks crowing, the muezzin calling – then going to our separate havens, and later joining up again to eat in the hotel’s fine restaurant, where our table was in a dappled orchard, or to swim in the pool.

We’d reached that stage of a holiday where you feel no need either to sightsee or to sunbathe, no need to do anything. We were in a beautiful place, our time was our own, and this was more than enough.

Antony Sher plays the title role in Kean, by Jean-Paul Sartre, currently running at the Apollo Theatre, Shaftesbury Avenue, W1

Travel brief: Abercrombie & Kent Private Travel (0845 618 2102, www.abercrombiekent.co.uk/private travel ) can tailor-make a 10-night trip from £4,900pp, with three nights in Istanbul, a three-night gulet cruise along the Lycian coast, and four nights in Datca. The price includes flights from London, accommodation and transfers. Or try Pettitts (01892 515966, www.pettitts.co.uk ) or Exclusive Escapes (020 8605 3500, www.exclusiveescapes.co.uk ).

Times recommends

• The HQ of the great morairaway.com villa scam: a postbox
• The bloggers' travel guide to. . . Copenhagen and Malmö
• A Devon holiday with mates - on a budget