As I drove southeast from central Athens, the hustle and bustle of Greece’s famously busy capital started to fade away. Not long before, I was shoulder to shoulder with packs of tourists, sweating my way along the pedestrian streets in the Plaka neighborhood, under the shadow of the Acropolis of Athens. But less than half an hour later, I was on a private peninsula — one fringed with white-sand beaches and covered with lines of green pine trees — getting ready to relax in luxury while surrounded on three sides by the sea.