I have of the two drawer Fye Boxes and I really like it. To me it is more convenient then the modern cordura packs. It's light, fits next to the body and does not get in the way of casting or moving around.
I like especially when I use big dry flies, there is plenty of room to not squish the hackle. Personally I would not be interested in the four drawer model as the do stick out to far. One of my fishing buddies has a three and it would be as far as I would want to go.
As far as the plastic I have seen them and I don't care for them.