KRASNAYA POLYANA, Russia — “What’s the score?” asks Svetlana Fedorenko as she enters a bathhouse in the Caucasus Mountains with her husband and friends: The U.S.-Russia hockey game was on, and most of the country was glued to a television.

A few miles away from the ski slopes of Krasnaya Polyana, where athletes are competing for Olympic medals, an outdoor bathhouse called British Banya is attracting visitors.