Media converters aren't really neccesary with any up to date switch that has a modular slot for some type of uplink media. Most nortel switches, and some cisco switches have this, allowing you to put in at least 1 fiber pair (tx/rx).
I would go with Justin or Servamatic's route, depending on cash availability. You don't really want to go above 300 feet with a Cat5e or Cat6 run. I'm sure it would work, but why risk it?
If you can honestly say that from switch connection to switch connection (patch cords INCLUDED) that it will be 270 feet, then go for it. Even 280, 290, or 300, but I wouldn't go above that. Spec is a little higher I think, 320 something? I can't remember. I could be wrong.
I would certainly do copper if costs are important here. I would pull 2 cat5E or cat6 backbone cables (mind you, I say two just to have one for backup in case you have any problems, it's a real pain to refish that stuff in after it's all said and done). And I would also pull in at least 1 25 pair. Some digital phones in today's phone system world still use 2 pairs, which gives you a realisic capacity of only 12 sets. Depending on your future vision for this building, and also if any fax machines are involved, etc, I might pull a 50 pair (or two 25's) and a few network drops as well.
Just forward thinking.