Can I get 24V from ATX PSU using 12V and -12V cables?
It depends on the device. You get 24V but the -12V is usually low power. So you could get a reference voltage. You could use it for powering very low power devices. Check the current rating of the -12V line on your PSU.

EDIT:

If you want to use your ATX PSU to get 24V, you should get a DC-DC boost converter from 12V to 24V. They are cheap and powerful.