Are cats unclean, such that it is impermissible to keep them as pets in the home?

If you are certain that cat feces has come into contact with something, then it is ruled impure and must be purified with water. However, if you are unsure whether a place in the house has been contaminated by its feces, the default ruling is that the place is pure. No place in the house is ruled impure by its feces except with certainty.