This fix isn't exactly elegant. And it's far from guaranteed to work, since only a handful of people have had success with it. But if your Xbox One has a busted disc drive, and you want to try fixing it in the most awesome way possible, flip it over and punch it. Hard.
There have been a few reports of this method working, the best one summed up in Dave D's video below, which theorises that the culprit is a gear in the loading mechanism that can be "realigned" with a little force.
And hey, as you can see in the vid, it worked for Dave. Though remember, it may not/probably won't work for you!