Iyyar 25, 5771
Is there a problem for a man to go to a mixed gym, very big facility at early hours when very few people are there
There could be two problems with using a mixed gym. The first is having to see inappropriately clothed women. If you are using a walking machine, for example, and all around you, and repeated endlessly in the mirrors, are women in sports clothes, it is hard to imagine that one would not have to see them. When this happens on the street, one can look away - and also there it is unavoidable. But here in the gym, it is both extended looking, and it is avoidable, as you are not forced to go in there at all. This problem may well be solved by using the gym at hours where, as you say, very few people are there, and you could find a spot where you could avoid looking at the women.
The second problem is the obligation to act in a fashion that lives up to people's expectations of what a good religious Jew should be doing. Generally religious Jews use separate gyms for men and women (for the reason we wrote above), and even if you use a mixed gym in a halachically acceptable fashion, if others could think you were doing the wrong thing, it would be forbidden. This law depends both on who you are, and what the situation is where you live. The more people expect of you ,the more problematic it is (for example. if you are the Chief Rabbi of Israel - don't use a mixed gym, ever!).
In summary - even though there may be ways for this to work, if a separate gym can be found it should be used. If not, then one needs to weigh up the above considerations and decide in each case depending on the situation.
I want to ask a question related to this answer
Yeshiva.org.ilThe Torah World Gateway