The first thing you're likely to see when you walk in the door is a wall of small lockers. These are for your shoes, so slip them inside and take your locker key back to the front desk. Pay the entry fee (the price should be posted at the reception desk) and hand over your shoe...
Starfield's mechs are little more than gradually rusting scrap piles or untouchable museum antiques. There's never a chance to pilot a mech, or even to fight against one on foot. Per the in-game lore, there's a reason for that: the construction and use of mechs has been banned ...