Ok, so you have a bunch of hats you don't want to through away because that would just be far too painful. I have a very simple solution for you. There's actually two versions of this and you can decide which is better for you, but they both work equally as well depending on what you're going for. What I mean is there is the conservative "out of the way approach" and then there's the decorative "look at my hat collection" approach. I have used both.

When I was a bit younger and didn't have the luxury of too much space for all my hats I used the decorative approach. Which was great because I was and still am proud of my cool and eclectic hats. I had tons of baseball hats, a few bowler hats, a cowboy hat and some random funky hats that were just fun to look at. I had all these hats and a small room. I used them to my advantage.

The Decorative Approach: Its simple. Grab you hammer and some nails, and making sure you are keeping a level line, hammer one nail for each hat on the highest part of the wall all the way around the room. This puts your hats on full display and keeps them out of the way without damaging them by cramming them all in some dusty trunk or box.

The Conservative Approach: Also Simple. Grab your hammer and some nails and do the same thing but in your closet. You will probably have to use multiple closets unless you have a big walk in closet. If you're married this is more then likely the method for you. It's a rare find to marry some that feels as strongly about your mangy old hats as you do so until you convince your better half that you absolutely need a sports den - you will be forced to use the closet. No big deal. You'll survive.