Personally wouldn't try this with a fire arm other than perhaps a .22 CB - which is about as loud and powerful as an air rifle.
But this would be a great set up for an air rifle.
I used to have something similar for BBs. Only the back stop was a couple sheets of old jeans. Using a card board box, you could shoot into box, the BBs go through the paper target, hit the jeans, then fall to the bottom of the box. Then you can just rather up and reuse the BBs.
ETA - just a note prompted by another post, BBs usually aren't made of lead, but steel, some times copper coated. Made gathering them up with a magnet easy.