ATHENS—For out-of-work Greek advertising executive Giorgos Fragioudakis, the writing was on the wall. His company had failed in Greece's economic crisis and he was out of a job.

So, with no income and with a family to support, Mr. Fragioudakis this year changed professions and tapped into one of the few growth segments left in the Greek economy: He became a souvlaki chef.

Business...