Revealing mistake: When Hawkeye walks into the mess tent naked as per his bet with Trapper, in some shots, you can see that he is really still wearing shorts. (00:07:40)
Continuity mistake: In some shots, Radar's exam questions are in his hand, in others they are tucked under his arm. (00:10:30)
Continuity mistake: When Frank is getting drunk he spills gin down the front of his vest. A few shots later, just before he collapses, it suddenly dries out significantly. (00:13:20)
Other mistake: Near the end of the episode Hawkeye is finishing his letter to his father. He writes "kiss mom and sis", but later in the series we find out that his mother died when he was a kid and he doesn't have any siblings.
Audio problem: At the end, while Radar is playing drums, he is using drumsticks but drum brushes are heard.
Continuity mistake: During the charity no-talent night, when the curtain opens on the Swinging Surgeons we can see Radar using a set of brushes on the drums, but in Radar's closeups he's using drum sticks, then in the wideshot it's back to the brushes again.
Continuity mistake: Casey's glasses go from dark framed to metal framed and back as he says goodbye to Hawkeye.
Continuity mistake: At the end, Casey comes over to say good bye to Hawkeye. As he stands there, the glasses he wears go from round, speckled-brown frames to silver oval, and back.
Continuity mistake: When the Swinging Surgeons are performing during the charity no-talent night, Radar's drum kit is set up differently between the wideshots and closeups.
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