I always figured it was just a tongue-in-cheek purposeful corruption of the 1940s FUBAR military jargon, but admittedly I never went searching for the origin.
FOO came first in the 1930s; it was pervasively used by the Smokey Stover comic strip. Then, students reinterpreted german furchtbar (awful) as the spoken term FOO-BAR. That became the backronym FUBAR.