Continuity mistake: As Charles talks to Finch, he hangs up the clipboard, and puts the pencil in his pocket. Then as he turns Finch to walk out, he puts the pencil in his pocket again.
Revealing mistake: In the closing scene, as the camera looks at the collected crowd, the camo net over the building poorly hides the edge of the "sky" background and the sound stage walls.
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