Just to be clear, if the peak is over the 12' top plate, I think 4' is the minimum. The bulk of any large items are going to go in the center of your loft. What you might want to consider is the mini-barn roof. This roof will give you about 15% more space in the same area AND you may have to raise the peak a bit. Check out any of the shed-related WEB sites for the mini-barn roof here.
I can suggest one thing. I have a 16' wide x 12' deep mini-barn with a loft with 8' walls, and 12' 6" peak height. I've suspended the loft using angle brackets instead of putting it on posts. This leaves the bottom open and if I have something that's 12' long, I can unbolt the bracket from the loft (not the ceiling), swing it out of the way, and put it up there (then re-attach the bracket). I build it so there are only 2 brackets (at about the 4' point) so that only 1 bracket has to be moved to get something up there. When I sided my house, I had some left over 12' sheets and channeling. By simply swinging only 1 of the brackets out of the way, I was able to get them up there. This is just a thought. I don't know if you plan on putting up a loft on the 'short' sides and/or on the 'long' side but either way, this may be something to consider. In my case, when you walk in the double doors on the 16' side, there's a 16' x 3' loft directly in front of you and an 8' x 4' one on the left. (it's L-shaped.)