My xbox turns on but turns off at random times

There could be a few things, like auto off is on. But I doubt that.

It could be over heating. I would try cleaning it personally. Though I'm convinced the insides are made of razor blades. Be careful not to misplace anything, and don't be rough. Though if you take it apart to clean and remove the dust, that's all on you and I'm not responsible in anyway. Also if it's under warranty still DO NOT OPEN IT. Just try blowing out the vents with compressed air perhaps, though that might lodge the dust further.

Also make sure it's not close to the ground, and not inside any cabinets or what not with poor ventilation.

It may also be the power source, inspect the cord for damage. If you can test it, test it.

Also I know my PlayStation did this, but it was due to the hardrive, so I doubt it's that.

Also, is there any other symptoms?

Sorry about your problem,

-Liz

Here's the Microsoft teardowns

Microsoft Game Console