I've seen this before on some devices, usually it is a burnt out drive motor. but to be sure when you power up listen to see if you can hear the drive, since it will stop spinning when you open the cover.

I found an old Gamecube at a second-hand store going really cheap. The catch was, the disc would spin up, and then stop again (with no disc message/indication).

If the drive detects no disc, it won't spin the motor. So if your laser is dying, and it cant see anything, it wont spin the disc.

So it is possible that the laser is dying/almost dead, but there ARE guides out there that may help with this; just google "gamecube laser calibration" and hit the first link.

Indeed, I had the same problem with my GameCube. The laser module died, but the spindle motor and laser carriage worm drive motor were both fine, as I could hear it both spinning briefly and trying to seek, but finding nothing. Maybe your lens is just really dirty? Try cleaning it with alcohol.