Revealing mistake: When Radar carries the field phone from Col. Potter's desk over to Hawkeye who is squatting in the corner, the phone doesn't have a phone line attached. It is an EE-8 Military Field Phone, which is a wired device, not a radio - even if Radar elsewhere says he has somebody "on the radio" when in fact he is using an EE-8. (00:17:35)
Other mistake: When the Colonel Griswald of the 1st Cavalry goes to leave camp, the Jeep he gets in is marked "4077 MASH" on the bumper instead of 1st Cav.
Continuity mistake: Before The Sniper starts firing, Hawkeye is carrying the two bottles of scotch firmly gripped by the necks. A second later, he's barely holding them by the caps, in preparation to drop them.
Other mistake: As Burns, driving the tank, runs over the Swamp, there's no sign of the still anywhere.
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