So, I did something stupid. I was editing my /etc/passwd file when I accidentally inserted “false” right into the name of the root user. So my root user failed to exist. Fortunately for me, I was in the process of retiring that server anyway, so I moved up my timetable and shut the old one down. I did, however, discover that the Linode Recovery Image works extremely well. I was able to boot into recovery, fix my error, and boot my old server up without a hitch, and all in the span of about ten minutes. I also checked out the Linode Backup process. It works exactly like one would expect it to.
All in all, I walked away from this minor catastrophe with zero down time and and as an extremely pleased Linode customer.