I use a slightly older Mac Mini as the target for my backups and due to it’s 1GB RAM limit, it cannot be upgraded past Snow Leopard, OS X 10.6.8.
One of the Apple software updates in 2012/13 introduced a bug for this version of the OS where certain mounted volumes are automatically self-eject after 10 minutes or so. Despite tons of Google search results on the subject, it’s not easy to fix the problem.
I send my Time Machine backups to a dedicated TimeCapule disk, then mount the TC on the Mini and use Crashplan to sync the backups to the web, along with a few static files/folders.
Once the self-ejecting problem appeared my backup redundancy solution no longer worked. Seeing so many HDs die without warning over the years, I progressively became more concerned about finding a workaround to the problem.
What eventually worked was to mount the TC over SMB instead of the default Apple Filing Protocol (afp) formerly known as AppleTalk. I first had to resolve the TC to an IP using the admin tools on my router, then was able to mount it with SMB using the Finder’s ‘connect to server’ option, illustrated here: http://take.ms/cv3Zw