There's a place almost exactly like the one the OP describes in NYC called the Russian Baths on 10th Street. It's been around since 1898 and also includes a small restaurant in the front (the only part where "clothes", usually a pair of the baggy gym short they provide) are required) that serves small plates of Russian food and vodka.
They also, in addition to the sauna, steam, 'nuclear' steam and weight rooms, provide different massage services. Nothing sexual goes on although it is incredibly sensual. It's very much primarily straight and mixed in age (although skewed older) but there are obviously gay guys as well. What's cool about it is it's just men being men, comfortably, together. No one cares who you are; everyone's just cool about it.