One day I tried to run the Xbox 360 and I see the well-known "RROD" (the Red Ring Of Death), the red ring that formed by the Xbox360 leds, and it's the worst that can happen to your console, is DEAD. Basically, I had 3 LEDs, ie "General Hardware Failure", which explains nothing.
Generally, the list of problems that Xbox 360 can present is huge. A good presentation is here. Based on this list, we can know about where to look.
The good thing is that the console can be repaired. We only need the necessary screwdrivers, heat-resistant paste (like the one we put on the PC processors) and 1 or 2 coolers, which you can already get from an old PC. Although there are many solutions, this is the most cheap easy way.
Open the Xbox 360 and take out the motherboard. You can find some great tutorials on the internet on how to do this.
Now you have to remove those big metallic "X" before continue.
You need to clean the processors from old pastes that are likely to be left with acetone. When you finish cleaning, put the new heat paste over again and put the heat sink again. I chose to put other screws, to keep heatsink tight.
Before close the console, you can make a final check. Connect the Xbox to the power supply to see if the coolers are warming up properly. They should be (almost) directly warm to make sure they are properly applied to the processors before proceeding.
Everything works fine.