CrossFit/martial arts gyms that allow access between classes?

I’m looking for a basic gym that has a few racks, a proper chin up bar, and space on the floor for olympic lifts and calisthenics. Not a sterile, corporate gym full of machines where no chalk is allowed.

I tried the CrossFit class thing and it just wasn’t for me, but that’s the kind of gym that would probably appeal to me. Are there any CrossFit (or maybe even martial arts) gyms in town which allow access to work out between classes?

Bonus if the gym is somewhat near the downtown core. I know Training Grounds may be the answer, but I’m hoping for something closer.