Voltage Drop and Safe Current Load on CAT5 Cable

According to a reference from the Handbook of Electronic Tables and Formulas for American Wire Gauge a conservative estimate for 24AWG power transmission is 0.577 amps. Current-wise, I'd expect you to be fine. I'd still recommend testing it and monitoring the temperature for a while to make sure. If you've got that cable in a tight bundle of some kind, it will get warmer than if it was in free space. On the bright side, wire that small won't take long to reach its ultimate temperature!

As for voltage drop, according to this AWG table, 60 meters of 24AWG has a resistance of about 5 ohms. 500 mA will drop 2.5V over that distance. You also have to consider that the current has a return path, so that's 2.5V drop in each of the positive and negative legs, 5V drop total.

If, on the other hand, the draw is on the low end of your spec, you'll only get 3.5V drop total. That means that the voltage on your load (the Raspberry Pi) will vary by 1.5V depending on how much current it's drawing. That's quite some variation, and you'll need to make sure it can handle that.

If it was me, I'd come up with a different plan. You're running really close to the edge of what's reasonable.

I can't comment on your answer but have to give you an advice, hope somebody turns it into a comment. When powering things over long lines, it's best to put more volts into it and put a DC-DC at the end where the device is, setting it to 5 volts (or having a fixed-voltage DC-DC). LM2976 or XL6009 converters are the best bet, since they're switching-mode and thus much more efficient. In the end, you have less current running through your wires and constant voltage powering the device, with no need to worry about all the things that could happen be the current high or voltage unstable. I have a couple of routers powered like that and currently am in the process of making a big system with 30 devices powered like this =) I'm sure that's your best bet in this situation.