My headless server running Ubuntu 12.04 stopped responding to ping,
and I could use some help diagnosing the problem. Here is what I know
so far.

The first thing I did was plug it into a keyboard and monitor and try
to boot it up. It gets to the GRUB kernel selection screen and picks
the first kernel on the list automatically. After displaying the
Ubuntu boot splash screen for a few seconds, it drops me to an
(initramfs) prompt.

A bit about the HDDs: this machine contains 4 2TB hard drives
connected via SATA. Each hard drive has a 3GB partition at the
beginning, a 25GB partition following that, and a large partition
taking up the rest of the device. The 3GB partitions on each drive are
in a software RAID 1 configuration (/dev/md0) which was used as the
swap device. The 25GB partitions on each drive are in a software RAID
1 configuration (/dev/md1) which was used as the root filesystem. The
large partitions on each drive are in a software RAID 6 configuration
(/dev/md2) which was mounted on /srv and used for user data.

Checking dmesg from the (initramfs) prompt showed that all of the
partitions on one device had been "kicked" from their arrays because
the partitions were not "fresh". My first suspicion was that this
drive had died, but googling this error message turned up the fact
that this can happen after an unclean shutdown and can be fixed by
running

# mdadm /dev/md0 --fail /dev/sda2
# mdadm /dev/md0 --remove /dev/sda2
# mdadm /dev/md0 --add /dev/sda2

I ran these three commands, and found that the first two did not
complete because /dev/sda2 had already been removed by the array.
Perhaps this is what "kicked" meant? The third command succeeded and
/dev/md0 was able to resilver quickly. I did the same thing with
/dev/md1 by adding /dev/sda3 again. Checking /proc/mdstat showed that
both arrays had recovered. I decided to leave /dev/md2 for later.

I tried rebooting, but it got stuck at the (initramfs) prompt again.
Going back to Google, I found that a system stopping at the
(initramfs) prompt can indicate a problem with the root filesystem, so
I booted from a SysRescue USB flash drive and ran fsck on /dev/md1. It
indicated that the journal needed to be replayed on the ext4
filesystem and did that for me. I tried rebooting again, but ran into
the same problem. Then I booted back into the rescue environment and
ran fsck -f /dev/md1. It checked the filesystem more thoroughly, but
that didn't end up helping either.

Just now I tried booting from one of the slightly older kernels in the
GRUB list, but I got the same result.