Pike suffered injuries on a cadet vessel that was an "old Class J Starship" in "The Menagerie". That would indicate there must be at least one other class of starship other than the one the Enterprise belongs to. Well unless the Enterprise was also an "old class J" which I doubt.
I think Class J is just Starfleet slang for "junk."
So it could be a starship much like the Enterprise, but older and going out of commission.
My theory is supported by Harry Mudd flying "a small Class J cargo ship." Spock had to specify small and cargo, because "Class J" only told you it was a piece of junk in Spock's estimation.