Cutting ceiling joist for attic ladder install
Yes, you can typically cut through joists, but you must make some additions known as trimmers and headers to the framing as well. Like this... 



Note that the trimmers run the whole length of the joists in place.

The ceiling joists will not hold much weight being only 2x4’s. I would not expect to be able to install plywood and flooring and walk around up there, but you could possibly store some lightweight things at the edges of the space and over the wall below.