You can ship pretty much anything -- maybe not by driving up to a UPS store with it, but through a shipping or moving company. Most of those companies don't give "generic" estimates -- they want details (what, where, when). Are you talking about a mattress, boxspring and/or frame or a piece of furniture? You can purchase a mattress online and most likely get it shipped for free, and chances are that will end up being a lot cheaper than shipping an existing mattress..
To answer your other question, the dimensions of a full size mattress are 54" wide by 75" long -- the height/depth can vary greatly (approx. 8-16"). That's without any packaging.