How long a run of wire are you talking about? With "long" runs, the concern would be with voltage drop (and secondarily with heating), but typical hobby installations that I have seen only use a few feet of cable to the motors.
The current carrying capacity tables do make some conservative assumptions which might not be relevant for the application that you are considering.
If I were hooking up those motors, I would probably use #20 or #18 wire. In fact, I just ordered four of them from Keling, and that is what I expect to be using. I personally would avoid using #24 wire for that application, although for short runs you could probably get away with it without ill effect. Since there are four pairs of #24 inside a cat5 cable, and the 381 oz-in motors have four wires, you could double-up the conductors.
The problem that I have with using typical cat5 cabling intended for in-wall (or in-plenum) use is that it uses solid copper conductors, which have a nasty tendency to break if the cable end is subject to vibration or movement. Some cat5 cables (such as those used in patch cords) use stranded wire which is less of a worry in that regard.