Can you have small eyes because you are born with cataracts?

Sometimes. The condition of nanophthalmos or microphthalmos are developmental problems with the eyes and can be associated with congenital cataracts. These are best detected with a comprehensive eye examination since glaucoma can be associated as well.
Other way around. Some patients with very small eyes (nanophthalmos) also have cataracts. Cataracts do not cause or lead to small eyes, however. Also, eyes may appear large or small even if they are not, depending on how deep the eye socket is as well as the size of the eye.