Don't all XBOX Ones come with a 1-3 year warranty? Call Microsoft up to replace it.
The Xbox one comes with a 1 year warranty, if it's a US based one, they would have about a month left......
It does sound more like a fan based issue, than a hard drive one....but, very hard to tell on that video...
To the OP, if you got your Xbox one on ebay or something, you could say that you got it as a Xmas gift or something. As this would not be fraud unless it was opened, as all Xbox one systems in the US are still under warranty for at least a month (system has a 1 year warranty and it has not been out a full year yet).
I would try to call MS first before trying to open it...If it IS a hard drive, a 500gb one to replace it will still cost $100 but, you cant just dump a drive in the system, it needs to be formatted a special way with the serial number and that is something Microsoft can only do...
People are putting larger drives in the Xbox one BUT, in the process they clone the original drive with the extra space.