Hi everyone - I've got a big problem and desperately am in need of help. I hope someone here can assist me for I am pretty clueless about hardware.

I moved about two months ago into temporary housing, so my desktop had to be boxed up and put into storage from May until last week. When I turned it off for the last time I shut it down properly (in case this matters).

Last week I moved into my new apartment, un-boxed my computer and to my horror, the computer was dead. When I plug the PSU into the wall outlet, the green light on the outside of the PSU to indicate if it is receiving power DOES light up green. However, the instant I press the power button to turn on the computer, the light on the PSU turns off and nothing happens. No start-up. No nothing.

I opened up the power supply and accidentally gave myself the shock of a lifetime, so the power supply is definitely seems to be receiving power.

Any advice on what this issue could be? I always kept my computer in perfect condition. Never got any viruses, don't download illegally, etc. I never had a problem with this PC over the past three years until this point. I assume it has something to do with the computer sitting in a box in storage for over two months, but who knows? Perhaps one of you do. Please help, any advice you can give will be GREATLY appreciated!

replacing the psu is only way to tell for sure, but i reckon it most likely is the psu.
They can be funny things can PSUs, it can work for years, and then when disconnected from the mains power, it might never work again when reconnected.
It's like they can be deteriorating inside the components, but as long as kept plugged in, they just hang on in there, but once unplugged they just give up.

It isn't uncommon for something such as the graphics card to become slightly dislodged when a system is moved. Before dashing out to get replacement parts that might not be needed, there is something you can try.
Disconnect the power.
Open the case. Carefully re-seat each card and cable. Are you at all familiar with the internal components of a PC? Do you know how to remove/re-insert the RAM? Any AGP/PCI cards. They are retained by a screw in the case and push into the motherboard slot. Remove them and re-insert.
Remove the power & data cables from the drives and re-insert them. Disconnect/re-connect the motherboard power cables.

Do all these things one at a time to make sure they are correctly re-attached.

Then plug back in the power and see if there has been any change to your startup problem.

Goin' Fishin' (Some day)

"I opened up the power supply and accidentally gave myself the shock of a lifetime, so the power supply is definitely seems to be receiving power"

To be able to open the PSU, it usually has to be removed from the case 1st...is that what you did? And then you plugged it back in before you opened it? Even if it wasn't plugged in, the capacitors can hold a charge for quite a long time. You're lucky you didn't get hurt.

Take Richard's advice & look for loose connections within the case, but make sure the power cord is unplugged 1st!

the reason I reckon it is more likely the PSU, is because the LED on the psu goes off when the power button on PC is pressed and it appears totally dead (which i take to mean no beeps, no fans etc)

it's probably not this, but it's a quick, cheap thing to test: try another power outlet.

If the PSU is receiving no power from the outlet, pressing the power button on the PC usually discharges any residual charge, which might explain why the green light goes out.

I agree with re-seating connectors. Connectors tend to corrode when power is removed from electronic devices for an extended time. With constant use the flow of electrons through the connectors tends to prevent corrosion.

An update - I bought a new power supply and I BELIEVE (not positive, but it seems correct) I installed it correctly. Still no luck. That mean I'm out a PC...?

The plot SERIOUSLY thickens: So I took the new Power Supply out, and repackaged it so I could return it. I put the old power supply back in, conected everything and just set it aside, I gave up. 5 minutes later I saw the power supply light on and figured what the hell, and to my surprise, IT BOOTED! Everything booted fine, everything lit up, etc. However, since I had disconnected it from everything, it was not connected to a monitor, mouse, etc. Everything was in my other room. So, naturally I had to unplug it and bring it in there. Clearly, I did not shut it down properly, but I had no choice. I set everything up in my room, plugged it in, power supply light went on, I hit the on button and then bam. Nothing. No power. ARGH! Help!