my shop is pretty expansive it covers the entire bottom floor of the house but that I did was cover the main area with a remnant I got from Home Depot and then put a old area rug down just to make it comfortable on the feet it is low pile so if screws or any parts fall on it there easy to find notice the red throw rug its just a beater so if I spill some glue or whatever its no big deal.
I also have a painting and mechanical area that I leave bare concrete so it is easier to sweep up but in your case just leave the carpet down if your renting the remnants at the big box stores are fairly inexpensive just to cover your space then like others have noted parts don't bounce off to neverland like they tend to do on hard surfaces.