Yes, they did. Here is a thread from two months ago where Rome, in particular, was covered.

I made a post about this recently. Broadly speaking, in the classical Mediterranean there were places to get food, but these were probably quick and cheap rather than cloth napkin sorts of places.

The earliest restaurants I know of are from the Sing dynasty, but these probably evolved from entertainment districts in the Tang so there may be precursors.

I circulated a link to this question on Twitter, and historian Carly Silver suggested Shopping in Ancient Rome: The Retail Trade in The Late Republic and the Principate by Claire Holleran if you're interested in that particular period / part of the world. The use of the search term "tabernae" also might help if you're not finding results on "restaurants" (though I agree that a non-Eurocentric view should be kept in mind if you're thinking generally).