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First, you should have backups. Why not get an external USB drive? You
can get those quite inexpensive. Heck, even a 64GB thumb drive
shouldn't be too expensive, though flash is not the most reliable.
No matter what, I would start with a good backup. Playing with mdadm
is relatively safe, but if it's your first time playing with it, you
can get yourself into trouble.
Hardware RAID controllers often auto-use replacement drives to rebuild
a degraded array, but mdadm does not. How exactly you go about
repairing the array depends on your specific configuration and which
drive failed. You will need to start by partitioning the replacement
drive such that it matches the existing drives (each array has to have
a partition equal to or larger than the same partitions on the other
drives).
Once partitioned (and rebooted, if needed), you will add the new
partition to the degraded array. Once added, mdadm will start
re-assembling the array. Note that if one of the arrays has /boot,
you'll probably need to run grub's install on the raw disk so that it
can be booted from.
The command to add the new partition to an existing array will look
something like this:
mdadm --manage /dev/md0 --add /dev/sda1
Obviously, the 'mdX' and 'sdYZ' will need to be adjusted for your
configuration.