APC has some UPS units (I have the 750W model) that will control some of its outlets based on whether or not the device plugged into a Master outlet is powered on (and using more than just standby juice). This might work, although these outlets are NOT battery-backed, so if power fails, they will shut off even though the computer will keep running.
An alternative, if speed is not required, is use 2.5" drives in USB enclosures. Many of these need no external power to operate.