Just do it. If you're afraid of autism procs, don't go out of your way to interact with people.
Few things to remember:
QUICK polite smile and a nod to staff members if they happen to look your way. Don't grin like an idiot or make excessive eye-contact, just jerk the corners of your mouth up a bit, nod and carry on doing your thing.
If you must interact with people, here's examples:
Someone is sort of hovering near the equipment you want to use but not obviously using them:
"Excuse me, are you using this?"
>Yes
"Okay, how long do you think you'll need it?" / "Okay, sorry to bother you."
>No
"Okay, thank you."
Try to make sure you'll never get asked this question if you're using the equipment. You might be asked how long you're going to use something, just give a rough estimate, no need to make it into a discussion. If you're asked while you're not using some equipment, answer "No, go right ahead." and make some space ( and stop hovering near equipment you're not using, it's annoying as hell ).
You need a spotter:
You don't, just don't be a retard with the Bench, or if the gym is amazing, the bench maybe has safety rails anyway. If you insist:
"Excuse me, could you spot me?"
>yes
"Thank you" ( and hope they know how )
>no
"Alright, sorry to bother you."
IF someone comes to actually strike up a conversation with you and you're truly so autistic you can't work with that at all, tell them that you're not very good at small talk ( inb4 "REEEE BETA" ). Unless they are massive assholes, which to be honest, is extremely rare, they'll be understanding. Owning up to your weaknesses is the first step to overcoming them.