I relate a lot OP's situation but every time I think about trying to talk to someone else I just get worried that if it does go "bad" (i.e. very awkward) then it will become to mentally hard to stay at the gym for the rest of the session or even come back and since its a place I actually like being in I end up never trying anything in fear of ruining a place I like. I don't know if anyone else feels like that but I just felt like leaving this comment.

As long as you don't bring up politics, religion, or money, you would be hard-pressed to make it so bad you wouldn't be able to stay or even come back. If things are SUPER awkward, just move to the other side of the gym or go to the bathroom for a little bit (~5-10m or so) then feel free to return, just don't talk to that person again other than a simple "hi" if warranted.

It’s probably mostly your social anxiety speaking there. If you go in not expecting anything in return it won’t get that awkward just go alright I’ll let you get back to it and go back to doing your thing.

I had like 20 years of social anxiety and it’s actually very anti-climactic when you can have a normal short conversation with a stranger. Not dramatic and no one’s traumatized

It’s a similar thing for me at coffee-shops that I frequent quite often and where I often see recurring faces. In a way it’s natural, “making contact”, so to speak, would only help transform those third spaces into potential-friends spaces, which might be good for some, but which might also seem less desirable for others.