External backup drive unmounted, then endless churn
I've had a similar problem with a 2 TB Fantom drive.  Worked flawlessly for years, then it started to spontaneously unmount for no apparent reason. Time Machine being on or off doesn't have an effect.

It had been chained through another USB device (a MiniStack drive), and simply by taking it off the chain and plugging it directly into the Mac, I solved the problem.

Before ditching the drive, try changing USB cables and changing the routing (even a different USB port on the Mac.)  Five years is a bit old for a hard drive, but it's not unreasonable.