Agia Eleni beach on Kefalonia, the largest Greek island in the Ionian Sea.

The journey from Albania to Kefalonia, the largest Greek island in the Ionian Sea, was long but easy. Each bus, taxi and ferry deposited me at just the right place and just the right time to catch the next bus, taxi or ferry, and when I arrived at midnight in Argostoli, Kefalonia’s capital, I simply walked down the street from the bus station until I saw a blue neon sign that said Hotel Tourist.

“You’re lucky we have a room,” said Pete, the friendly overnight clerk. “We’re the cheapest hotel in town.”

That room was 42 euros a day, about $55 at $1.28 to the euro, and as generic as the harborside hotel’s name, yet I had a small balcony, and the sheets were crisper than any I’ve slept on in two months.

But when I woke up in those crisp sheets, I felt worse than I have in two years. My chest was heavy, my limbs weak and achy, my hacking cough painful and unrelenting. I had a cold — a bad one. My luck had run out.

Readers of this column may know that I’ve had phenomenal good fortune so far. The places I’ve seen, the people I’ve met, the very fact that I’ve been able to undertake this adventure, all have had as much to do with luck as with any Frugal Traveler skills I’ve deployed.

Even my selection of Kefalonia (also spelled Cephalonia) was the result of serendipitous factors. I had it in mind when I asked readers for Greek island ideas. I liked its proximity both to Albania and to Ithaca, Odysseus’ home, and it seemed a good middle ground between too-chic-for-me Mykonos, overtouristed Corfu and far-flung, unpopulated rocks like Kasos. Plus, it had a smidgen of glamour — the 2001 film “Captain Corelli’s Mandolin” is set there, for the most part just before the 1953 earthquake that devastated the island. When several readers suggested I check it out — one called it “heaven on earth” — I thought: Kismet!

But whom the gods love, they destroy. For two days, I lay listlessly in my room, rousing the energy to get up only around noon, and by 1:30 the powerful summer heat would set in and most businesses would close until evening. And so I would return to Room 26, and sweatily doze till dark, afraid to turn on the air-conditioning, which I blamed for my sickness.

There are, however, worse places to fall ill: Cambodia, for example. Argostoli is a low-key town, with an unpretentious shopping street, Lithostroto, on which the young and the beautiful drink cocktails and coffee frappes in the evening before wandering over to the Central Square for Italian, or to the quayside for seafood. (Grilled Fish Index: 40 euros, or $52, per kilogram at the otherwise affordable Kiani Akti, 30-26710-26680.)

Alas, I could not join their fun. I had no appetite and had to force down every gyro and Greek salad. (Frugal Traveler tip: Illness helps trim food costs.) One night, I worked up enough energy to see “Good Night, and Good Luck” at Anny Cinema (Harakopou 54; 8 euros), an outdoor theater. The next, I sold five books for a euro apiece at the Bookmark (Lithostroto 4; 30-26710-27616; thebookmark-kefalonia.com), whose owner, Brian, a Northern Irish transplant, told me about the Kefalonia I had yet to explore: the trendy town of Fiscardo, whose posh Italian visitors don’t walk but glide along the harbor; Corelli’s Beach, now fenced off to visitors; and the island in winter, when the air takes on a lustrous clarity and hard rains turn the hills lush and green. He also let me read “Odysseus Unbound,” a 50-euro tome that posits that the Ithaca of “The Odyssey” is actually Kefalonia.

Cold or no cold, I had to see Kefalonia. With one day left before I was to fly to Istanbul, I rented a 50cc scooter from Sunbird (15 euros; www.sunbird.gr), declined the proffered helmet (sorry, Mom!) and set off to buy provisions. Then I discovered I’d lost my ATM card.

Down to my last 25 euros, I surveyed my options. A friend could send money via Western Union, or I could get a cash advance on my Capital One Visa card. The former would cost around $100; the latter, 3 percent of the total. I chose the latter, making my withdrawal from the National Bank of Greece, which had no qualms about completing a transaction for a noncustomer.

At long last, I mounted my MBK Ovetto and sped down the road, heading for the west coast, which the reader-recommended GreekTravel.com says has the best beaches. Within minutes, I’d escaped semiurban Argostoli and was deep in the Greek countryside. The dry rock mountains rose up all around, dotted with olive and orange trees and the occasional bulbous spires of churches. I swerved onto side roads that S-curved up and down steep slopes, my neck and arms baking in the sun, my hacking cough and rotten luck forgotten. I felt free.

Using my free tourist map, I made my way northwest at 22 miles an hour, the scooter’s top speed, to Agia Kiriaki beach, an isolated crescent of sand and pebbles lapped by blue Ionian water and occupied by just four people. I pulled on my goggles and jumped in, marveling at the skein of shadows cast upon the sandy floor; I hadn’t seen water this clear anywhere yet on my trip. Now I understood why people come to Greece.

But after sunning myself there an hour, I craved new turf (and surf). And I knew that on an island with 158 miles of coastline, every road would lead, eventually, to a beach.

After passing through the village of Damoulianata and hanging a right at the hilltop Sunset Taverna, I entered a land of gentler terraced slopes, olive groves and low stone walls. The empty road wound and wound, the sea teasingly close but always just out of reach.

But at last, I neared a cliff, and as I rode my brakes down yet more switchbacks — these steeper and not well paved — I was worried. The Ovetto had strained ascending even the easiest hills, and given how things had been going, I figured I stood a good chance of stranding myself.

I pulled the bike to a stop at the flat spot of one curve and dismounted. I descended a little to see what lay ahead, and then gasped. Below was a tiny white beach at the bottom of a looming, shadowy mountain; I knew in an instant I had found what I wanted. My luck had returned.

Agia Eleni is, to my mind, the perfect beach. Unadvertised, hard to reach and utterly pristine, it has a 20-foot cliff from which you can dive, a second, smaller shore reachable only by wading or swimming, and schools of fish that follow you as you swim among its rocks and over areas of undulating vegetation. And the people who go there (and who revealed its name) are just the sort you hope to see at a Greek beach, the kind who inspire comparisons to the deities of antiquity.

It was getting late, and I had a long ride back to Argostoli. The scooter wheezed up the road, reminding me that I myself hadn’t wheezed in hours. My cold, my money woes seemed impossibly distant then, mere trifles to be borne in stoic sorrow. My health will improve and a new ATM card will arrive by FedEx soon, but this day in Kefalonia was irreplaceable, the reward for yielding to the capricious follies of fate.