Get an archery based, foam target (or animal..!). You can get them in many different outside dimensions (and thicknesses).
I shoot both my crossbows and now my air guns indoors. Standing at the back door (could be closer, could be standing outdoors!), I shoot across the living room and down to the end of the hall, where a 16" thick, 18" x 18" square foam target, as it sits on a box for good height.
Stops ALL pellets (.177, and .22 anyway), no bouncing back on the floor, quiet, no ping when it hits (sorry if you like the extra noise !) It will easily catch .25 cal. pellets, and probably .30 cal, but probably not anything larger.
When the target face is full of holes, you turn it around to the new face and start over. The foam is "sort of" self healing. It will mostly close up around the pellets. Until you get several in the same location, then the hole will be a...hole. I've had this same target for years, with crossbow arrow holes in it, now pellet holes...and no sign of slowing down.
The actual target is paper with various targets printed from various places on the net. I just use an Allen wrench to hold the target to the foam. Punch the paper full of holes, install a fresh "target".
Simple, clean, not some nasty looking, made up backyard thing.
Mike
P.s. - This isn't the one that I have, but this is a good idea of what I have. Mine is thick like this to stop 400fps arrows. My next one will be thinner to stop just pellets