Visible crew/equipment: When Colonel Potter surprises everybody with his "little Olympics" idea in the mess tent, after Margaret comments, "I just know I'm not going to have anything left for Donald," at the start of the next shot the boom mic casts a very quick moving shadow on BJ.
Visible crew/equipment: While Colonel Potter has everyone gathered in the mess tent to surprise them with his "little Olympics" idea, the studio soundstage is visible at the top, left side of the screen.
Other mistake: When Penobscot is approaching the group, you can make out "4077 USA MASH" on the bumper, even though the the numbers and MASH are partially obscured.
Continuity mistake: As Ames is pitched the idea of diet and exercise, he has his fork up with a bite of egg on it. When Hawkeye reaches for the fork, it's upside down and clean.
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