It goes beyond just some doors -- there's several lines of dialogue in the movie ("Half the doors won't open," "The ship's in pieces"), plus Scotty having just "fixed [that damn thing]" (the Red Alert notification-system), the turbolifts messing up, and the transporters being completely down, which pretty much point to a massive systemic failure of some form aboard the Enterprise.
``None of the systems are working! Well, except for the warp drive. And the impulse drive. And life support. And shields. And photon torpedoes. Shuttlecraft. The brig. All right, nearly all
the systems are working, but darn it, the doors are glitchy! How can we get under way like that?"