You have to look at the spirit of the law more than just the letter of the law. If, by work out, you mean beat your brains out for physical and competitive reasons, I’d say no. If you mean to engage in exercise mostly for leisure, ie. stretching, running for fun or play, going to the gym on Sunday may be permitted if it is not for a long period of time. Some people cannot go at other times and do not have a suitable place to engage in physical activity for relaxation. Example, a parent who lives in a small studio apartment in a city takes his child to the gym for fun. Compare with a man who frequents the gym for the purposes of competitive body building, no. Let Sunday be a day of rest. I said man, but the same applies to women although in general, women tend to be less frequent gym-goers.
The same principle applies to non-servile activities. A person should not perform what he does for a living on Sunday unless it is out of necessity or charity demands it. Non-servile work can be just as tiring, even more tiring than servile work. If I write computer programs 40+ hours per week, I shouldn’t do it on Sunday. God did not make us to be able to work days on end without a break. Ask anyone who works rotating shifts or has to work seven days to pay the bills.
Sure, if we had truly Catholic states, these conditions wouldn’t exist and gyms would be closed or very limited in available activities. But we do not have Catholic governments and must do the best we can to honor God under actual, not fictitious circuмstances.