i bought 2 or 3 x360. the last i bought still going well! even after being abused (used as a stool, etc). The DVD Drive no longer can open without being punched/slapped tho.
So I was going to link to a youtube video showing you how to change your Xbox 360 drive belt with a pair of long tweezers or kelly clamps, but every video I saw have people disassembling the entire system. I can't be the only person who knows you can do this. It take a little trial and error the first time, but a coworker and myself used to see how fast we could do it and I think his record was something like 45 seconds. We charged $10 to do it, so it cost $800 a hour to do that repair at that rate. Anyway, here's how you do it.
You can buy a new belt for a few dollars on Ebay or Amazon, but you can also boil your existing belt and it usually works. Just put it in a coffee cup with about a cup of water and microwave it for about 2 minutes. Dry it of and it should be good. If it isn't, just do it again. Sometimes it takes a little longer. Sometimes if you order a belt from ebay or amazon, it will say that it's a die cut drive belt, and it's just a tangle free hair tie. Like the ones kids make bracelets out of on the looms that were all the rage a few years ago. So maybe just buy a pack of those at the dollar store in a pinch. Kelly clamps make it easier. Any hardware store, including discount ones like Harbor Freight or even many dollar stores have them. If you can't get them you need fairly long tweezers with good grip.
So, pop off the faceplate and remove your hard drive. Get your drive to eject. With the drive open, unplug the power. You don't want the drive to try to close when you start messing with the belt. There is a black drive belt near the middle that connects a larger pulley in the front to a motor pulley in the back. Grab the belt with the tweezers and pull it off. When the belt is off, be careful not to let the drive close because it will no longer eject, and you will have to do it manually, which can be a bit tricky. Boil your belt or get your new one. Put the belt on the motor side first (the smaller part in the back), and stretch it around the larger front pulley. That's it.
Pro-tip: Have your console flat at eye level. Don't set it on your lap pointed upwards. The door will close and it's a pain.
Edit
Found a video of a guy doing this with an OG Xbox, which is actually a little harder because you have less room. The removable faceplate on 360 makes it easier. He also stretches big pully to little, and I do find that easier on OG Xbox for whatever reason, but little to big on 360 is easier for me. YMMV.