A lot of these old city armory buildings (at least here in Canada) were built to be the headquarters of a regiment, drawn from members in that town/city. Unlike now where there are large bases, usually on the outskirts of towns or in remote areas. Furthermore, when you join the armed forces now, they post you to where they need your particular aptitude and skills. Contrast with how the Essex Kent Scottish were headquartered out of the old Chatham Ontario Armory (its now a banquet hall) - all the members hailed from Essex and Kent counties in southwestern Ontario.

As such (a headquarters for a regiment sized unit), the armory was used for a whole bunch of things. Training, recreation (think basketball and the like) and drills. Also for social events. Think of several hundred soldiers in full kit training and drilling in a low ceilinged space with officers and NCOs barking orders. It would be loud. It would be hot. Tall ceilings help with both of those.

While most vehicle maintenance would be performed in a motor pool garage in a separate building, the doors of the amory would be wide enough to fit the vehicles through if they needed to. The NCO who gave me a tour of the armory building above said the door would (barely) fit an M3 Sherman.