TIBERIAS, Israel—This ancient city on the Sea of Galilee was once a popular weekend beach getaway for Israelis and a magnet for foreign visitors drawn by its 2,000 years of history.

Now, seagulls and stray cats outnumber people on the city’s seaside promenade on the Sabbath, when booths selling cotton candy, popcorn and fake tattoos that residents say once attracted weekend crowds are closed.

Many...