An interesting question is raised by HBG's post and the construction picture above - Normally an elevator is in a double-drywalled-off shaft (1 layer of 5/8" 2-Hour Fire-Code Gypsum on the inside, and a second on the outside) that is deliberately blocked off from the rest of the building for fire resistance. But normal elevators don't have all the show elements in them either.
So.... How isolated are the Stretching Rooms from the upper floors on the Theme Building? I would expect the shafts are all drywalled-off to the rest of the building except for a bunch of access doors around the walls and on the ceiling (that have standard Elevator Door safety interlocks) to change out the Stretching Portraits and lights.
But they could also have it isolated off with Plywood or planks instead, and not worry about fire because "It's a Theatrical Set! And the building has Sprinklers!" which might have passed muster in 1964. I doubt any Elevator inspector would just let it all be exposed and let anyone get into the upper stories while the stretching rooms were in motion.
And knowing Cast Members, they would want to go up there and explore if they had a chance, even if the park is open and the Stretching Rooms in use. The '2nd Floor' space over the Foyer would make a nice spot for a break room, just like the 'Basement' space underneath it would be a logical place to put the elevator controls and pumps.
--<< Bruce >>--