They sell 3' x 36' rolls of cloth ground cover at OSH, so I'm going to make a trip to get a roll or two so that I can cover the entire floor with that. Then, I can put down gravel or bark or whatever else I feel like sticking down on the floor.
Been there, done that. Ground cloth is a reasonably good material for preventing growth from below, but the minute you get any accumulation of soil or organic matter on top, you will have weeds. Oxalis loves that environment, as does Cress. The only floor treatment that is guaranteed to stay clean and weed free is poured concrete, but I doubt you want to go that route.