If there's a way to restrict the memory used by the mzscheme process, you could set an upper limit, and then
(with (things nil n 0)
(prn (++ n))
(push (new-thing) things))) ; not tested!
and wait for it to die. You won't have the memory size in bytes of each thing, but at least you know how many fit in your process, and that might be the more important number depending on your purpose.