I can't speak for welding...nor is my floor exactly fancy.
I'm working on a friend's 1956 Olds...and it leaks. Alot. So, I bought that heavy plastic like you would use in construction. It comes in 10 foot wide rolls. (or about that width) I just roll some of this out and park the car above. When the sloppy job is done..I just ball it up and throw it out.
I have a bunch of old moving blankets that I throw on the floor, they take all the weld spatter and they don't catch fire *TOO* often, LOL. Someday I will get some "proper" welding blankets, but these are working fine for now.