my first thought was laminate if you wanted to put someting down. I personally have a bare concrete workshop floor in my shop while the rest of the basement is finished.

Ditch the shag.. The red hot piece of the screw you cut off with a dremel can make a nice fire in carpet. Especially after you get a few spots of after run oil on it. I would put down vinyl composition tile. Maybe do a nice checkerboard patterm. 1/sf for the tile if you look around.

You may ponder, but you won't get anything done.

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.

Mine has a bare concrete floor that I covered my working area with those rubber tiles that lock together. Got them from Harbor Freight for $9 a 4 tile package. Keeps the feet warm, easy to keep clean, HOWEVER, one word of caution when walking on them. Make sure there is no water under them! My basement leaks when it rains hard from a certain direction, poor seal between the old foundation and the new wall they put in for the addition where my shop is. I didn't notice that some water got under the tiles, the water leading to it dried. I took a nasty spill when the tile went one way, and I went the other. My shoulder it still bothering me from that.

I have since put down some textured floor paint so now the tile has something to grip to and wont slide.