HERAKLION, Greece—The first time he despaired of his debts, Vaggelis Petrakis drank a poisonous brew of beer and gasoline.

A note he left didn't mention the financial woes of his fruit and vegetable business, of which his family was well aware. Instead, he left instructions for his children on how to look after his animals. "Put mother rabbit in a different place from the little rabbits," the note began.

Then...