There may be some establishments selling food or alcohol anywhere and anytime there is currency or a medium of exchange to be used for food and alcohol, but you might be interested in the Ancient Greek and Roman thermopolium and taverna which is one of the earliest evidences we have of "fast-food".

The thermopolium was an Ancient Greek and Roman small restaurant bar selling food and drinks and sold ready-to-eat food for lower class and poor people who cannot afford a kitchen or a cooking oven in their houses. It is comparable to modern fast-food today in the sense that the food was ready-to-eat.

In the Agora of Athens, there was excavated an evidence of an ancient Greek restaurant, and there was also a nearby wine shop selling imported wine from throughout Greece.