This term goes way back into the sailing days. It basically means a tight confined space. This colorful phrase evokes strange images of feline cruelty. In fact it has nothing to do with cats, but the real story is at least as cruel.

The "cat" is a cat-of-nine-tails, a type of whip used to discipline sailors on old sailing ships. The cat-of-nine-tails has one handle to which is attached nine thin strips of leather, each perhaps three feet long. The cat-of-nine-tails would be used to administer lashings that would sting and leave welts on the recipient.

The whippings would take place on the deck, because below deck there was not enough ceiling height to swing a cat-of-nine-tails.
Interestingly, the Disney cartoon character Mickey Mouse once gave a graphic illustration with some comic relief to this phrase. In the 1920's "Steamboat Willie" was release as Mickey Mouse's first cartoon. There is a scene in the cartoon in which Mickey is on a boat. He proceeds to pick up a cat and swing it around by the tail. Apparently this particular boat had enough room to swing a cat.

So, if you work in a tight space and want to get this point across to your boss, or you just want to confuse him, tell him, “There isn’t enough room to swing a cat in my office.” Just make sure he isn’t a ship modeler before you spring that on him.