Call Nintendo. They probably still offer tech support, right? :)

@BrockNRolla:  HA! Yea that would probably be an interesting phone call.  Maybe it is still under warrenty......

I miss my N64 :(
 
Anyways, I have no idea, nothing like this ever happened to me.

I have no advice but good luck man, I know how painful it is to pull an old system out of storage expecting it to work and finding it busted. My original NES and SNES are both relegated to decorations right now, and I fear my PS2 may follow them soon. My N64 still works fine, though the 'party hub' nature of the system has not been kind to the controllers and I now only have one that works like it should.
 
If worst comes to worst and you can't buy a new system or fix your current one, it is quite possible to emulate N64 games. You don't even need a great computer to do it.

Download a N64 emulator and Roms of games that you own.

Do what this guy did

My SNES went through someone similar. All you had to do was touch the console anywhere and it would reset itself. The slightest vibration would cause it. Haha, we couldn't jump in celebration or we'd lose everything.

So i have some info that might help here. I recently bought a n64 and started playing majoras mask and after an hour or so the console reset itself a few times and then kept resetting every 2 or 3 minutes after that. At first this obviously seemed like an overheating issue since the time between resets was getting shorter and shorter. To test this I put in Perfect Dark and it played fine, which led me to believe that it was a problem with my nintendo brand expansion pack since I heard that Zelda games were more likely to have the reset issue (I assume this is due to the fact that only certain games utilize the expansion pack i.e. Perfect Dark only needs the expansion pack for multiplayer but zelda games will shut off the console if you remove them while playing). So I took the expansion pack out and the connectors were absolutely immaculate not a single piece of dust on them, but I decided to clean them anyway and now I have not had the issue since. So I guess the connectors on those cartridges need to be clean on a microscopic level! If my problem comes back I will post again to let u know if and how I fixed it.

sounds like its broken.

I got the exact same problem. It's overheating, the base of the system suffocates when it's over any surface (except the anti thermal ones I guess) because there's no decent airflow. The solution is very simple and you don't need to tear it apart or buy a new one. I found that you just need to get it off the ground with the help of some supports. Try to find some cilinders or any object that can make the N64 stand on top of them like the parthenon of Athens (in my case I used those Wii U supports that comes with the black one). The airflow will then function properly and the resetting will stop.

Hope this helps ;)

@koxen: Thanks for the heads up. My system isn't broken, but it might be in the future. Now I know.

They'll fix it in the next patch.