If it does nothing except spin up the fans, I'd bank on RAM or CPU being unseated. Also check that your heatsink is attached and the fan is working - many systems will shut down if there's a CPU fan fault or if they detect overheating so that the CPU isn't damaged. After that, try replacing the power supply. The fact that it's clearly doing something (fans / HDD spinning) leads me to suspect something else initially but certainly doesn't rule out a power supply issue. Check that the 12V (4/8 pin) cable is hooked up to the motherboard. If it's unhooked, you'll get exactly what you describe.
PCs and electronics in general do NOT need to be stored at "room temp", but it would be recommended if possible. They're usually good for storage over about 0-50C and potentially quite a bit wider (-20-70C is common). They can potentially survive freezing so long as there's no condensation. Higher temperatures can cause premature degradation of flash memory. You should ALWAYS let the device come to room temp before turning it on, though. This usually means 24 hours under controlled conditions. NEVER power up heat producing devices (CPUs, CRT monitors, etc.) from a very cold state. The thermal shock as they heat up rapidly will often break things.
Some precision instruments may be damaged or need realignment after being subject to temperature extremes, but such instruments usually need realignment every so often, anyway. PCs would not generally fall into this category with the possible exception of hard drives. Dirt, condensation, static discharge, etc. are all far more damaging to electronic systems than mild (i.e. 0-50C) temperature extremes. If you're even remotely comfortable in it, the electronics are fine with it if they're not in operation. While operating, it depends entirely on the system design. 25C is usually the design target.