Other mistake: When Hawkeye finds the New York Times Crossword he says, "1 down, four letters..." and "1 across...four letters..." and BJ's answers are "chap" and "cast" but in the shot of the newspaper puzzle we see that both "1 down" and "1 across" are actually five letters each, not four.
Revealing mistake: Klinger makes a big show of removing a big nut from the Jeep's leaf spring. The nut is clean and shiny whereas the rest of the undercarriage is painted and dirty. If you look closely, there are no screw threads where he takes it off either, there's another hex under it. The simple reason for this is that that nut doesn't belong there and never was affixed, Jamie Farr just pretended it was.
Continuity mistake: After Radar leaves the message with Stevens for Hawkeye's friend on the Essex, it cuts to Klinger removing a nut from the jeep and he's holding the wrench with his thumb at the top, but next shot he's holding the wrench the opposite way.
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