I attend an all-women’s gym in Jerusalem. The owner herself is frum and no men are allowed in during opening hours. Do I have to cover my hair there while I work out? My husband thinks that this is still considered a “public place” but I think it’s more like a “beit merchatz”. Clearly, I and other frum women who work out there wear short-sleeved shirts and pants which we would not wear in public.
From the various rabbonim that I have consulted on this matter, it seems controversial, just like the two sides of your question. Some said that it is not permitted and others said that it is. However everyone agreed that it isn’t the preferred thing to do, and if possible she should be dressed b’tznius