If you reformat the partition:
You will lose the data on it (since that perspective does not seem to bother
you, I guess you have a recent backup);
The problem should be fixed...
... but not for long (or at all) if the drive is actually defective!
You can see the SMART data of the drive. The easiest way is probably:
to install the "gnome-disk-utility" package in the live system;
to launch "Disks" from the "Control Center";
to click on the disk (in the left panel);
to click on "SMART Data & Self-Tests" in the "burger" menu (the three stacked
horizontal segments).
As the name says, you can test the disk from there too.
On GNOME Disks' main window, after clicking on "Partition 5", the button with
three gears allows you to "Format Partition...". You would then have to edit
accordingly /etc/fstab's line that refers to /home. I mean the file
/etc/fstab of the installed system (you can still write on its root
filesystem, right?). And you would have to recreate (e.g., with 'mkdir') the
home folders of the users, setting the same uid and gid (e.g., with 'chown')
as before: they can be respectively read in the third and fourth column of
/etc/passwd. Again: I mean the file /etc/passwd of the installed system.
Well, that instructive process or reinstalling everything. :-)
