That would cause any disk-write-dependent service to crash, yeah, but the kernel isn't one of those. Syslog will stop and a bunch of things will complain, but I've always been able to SSH in and free up some disk.
Did you read/listen to the story? A non-tech person called him and said "the server crashed." I am pretty sure they would say that in case of "any disk-write-dependent-services crashed, but you could ssh in and free up some disk." The criminal boss wasn't gonna "ssh in and free up some disk", that's why he had an IT guy!
Anyway none of this is particularly important for the story, but I don't think the guy telling it is lying for dramatic effect, I think he's probably being honest that the boss saying "the server crashed" made him suspicious cause that server never crashed, and I too found it amusing for this to be included in the story (as a sort of by-the-by advertisement for linux). (Also, though: it turned out the server really did "crash" in some way).
Non-IT people may say it's a crash no matter what actually happen. "Crashed" "Freezed" "app not working" "can't print" is the same thing for them often.
BTW I've seen an error where app server couldn't write files to a directory but only for specific filenames (that wasn't already created). Turns out if you have dozen thousands of files in one directory the hash table has collisions and some files you can create while some other names you cannot. It was lot of fun to discover that :)
And customers described it as "server doesn't work" but when we connected the randomly generated names it was trying to write were different and it worked.
