this actually just recently happened to my 360 and it was taken apart so i dont know if that had anything to do with it but it would spin the disk for a few seconds then stop. i actually bought a new xbox it was so bad. and i know this sounds stupid but open it up and just clean the disk drive mine was very dusty and i thought just getting rid of the dust would work but it didn't, i researched online for a LONG time and couldnt find anything then i learned how to clean the disk drive from top to bottom i spent about 20 min cleaning it, and now it works perfectly fine so give it a shot before buying a new 360 like me haha good luck!

I had this problem before and opened up it ended up being the little laser thing that scans and reads the disc it was loose and I just tightened it and bam.

try cleaning the laser with a q tip if you do open before anything