He reached his family’s old property, in a field with a view of the sea and snowcapped mountains, and stood above a shin-high rectangle of stone. It was all that remained of the home where his parents took refuge during the Second World War. The migrants had taken the wooden walls, the windows, the doors and the roof. “Everything,” he said with a shrug.

About 6,800 asylum seekers are jammed into the camp and fighting the elements in the olive groves and pine woods of the hill. Below is a quaint port town that is home to about 6,200 locals like Mr. Meletiou.

Together, the locals and asylum seekers bear the shared brunt of forces beyond their control — Greek government dysfunction, the cold shoulder of the European Union, the chaos in the Middle East and the geopolitical calculations of Turkey.

And many here fear that this verdant tourist destination — famous as the birthplace of the goddess Hera and the philosopher Epicurus — is a preview of the future if the Continent doesn’t get its act together.