We had a "computer room" (old office on the top floor, south facing, floor to ceiling windows and no AC).
You've almost described our server room at my last job - the one I rebuilt, wired properly, but wasn't given any budget for aircon (and we probably wouldn't have got permission from the landlord). We did have forced air cooling - one fan sucking "cool" air in, another blowing hot air out. Except that the inlet and outlet were above a tin roof - so in calm summer conditions, the combination of heating from the roof, and pooling of the hot air we'd blown oout meant that the intake air wasn't all that cool. I also estimated that we needed at least twice the airflow - which meant that as the temperatures went up, the servers sucked more air than we were supplying and so that sucked hot air from the back of the racks round to the front - increasing the temperatures further, so the servers ran up the fans more, so ...
One summer it got really hot, and we took a punt on a second hand "swamp cooler" (we call them evaporative coolers here in teh UK, but the merkin version sounds better in some ways). It sat in the front of our server room, and we filled it twice a day - IIRC it could get through something like 40-50 gallons some days !
It certainly did the trick, though the front inch or two of all the server metalwork went a little rusty.
And the kit (mostly) had to fend for itself over the weekends.