Revealing mistake: In the swamp during the movie, Frank comes up and talks to Igor, who is running the projector. Although there is a film running, there is no film on either the supply or take-up reels.
Factual error: In the beginning, Hawkeye says that the patient was full of shrapnel from a claymore. M*A*S*H was set in the early 50's, and the M18 Claymore was only in the development stages in 1952, and not accepted and battle used until Vietnam in 1961.
Answer: Even an Army field hospital would have SOME weapons for self-defense if the enemy attacked. What's meant is that it's not a combat outfit, and therefore wouldn't have enough weapons to actually attack anyone. (It's also mentioned a few times that the Officer of the Watch is supposed to carry a pistol at all times, but Hawkeye refuses to do it when he's in charge and Colonels Blake and Potter don't enforce the rule).
Captain Defenestrator