If you are going to use a piece of plywood as a backstop make sure to lean it forward (towards shooter) so BBs are redirected down into the ground. If it's vertical it will bounce BBs back at you (don't ask how I know) and leaning away from you will send them vertical. Even though they are BBs they can still do damage to people downrange.
Probably the best thing to do with limited space is a pellet trap. You can buy them, or just google pellet trap plans or homemade pellet trap for lots of ideas. Also, I wouldn't use plywood less than 1/2 inch. 3/4 inch would be even better. Even those small BBs can make it through 3/8s of an inch of plywood. Once again, don't ask...
