@sethturner50892 just like @andrewsawesome already said, time to take it apart. Yor GameCube could be over 20 years old and things happen when age creeps up. Use these guides [[Topic:Nintendo GameCube]] to work on it. Clean the fan, check the capacitors and check the traces on all boards, for corrosion. That is pretty common after that long time. If unclear or if anything looks odd, you can always post some images with your Question [guide|21499]
If you are still using the original power supply, then check the output voltage on it. Wiggle the cable and see if that turns off the output. Of course, you will need a multimeter for that.