A typical install would us 2x4's with plywood across. Basically mimic wood stud framing. Doing the same with 2x2's should be fine for this application. I'm assuming a 100lb dead load. Use 23/32" sheathing plywood or MDF. Mount with nuts/bolts into the plywood and framing nails elsewhere. Weak point would be tearout of a connector not deflection of the 2x2 beams. If you are paranoid you can use metal wood ties to enable bolts elsewhere rather than toenails. I think once you hang the 2x2s from floor to ceiling you will see they are plenty strong enough to hold a grown adult.
I included a picture that I think represents what you are trying to do except behind this plywood/frame would be a basement wall with pipes you don't want to attach to.