You will not be sorry about purchasing the Accuquilt Studio. I have one and it does make cutting go faster and the cuts are much straighter (for me) than my own cutting skills allowed.
I store my dies in a china closet. The big ones just make it inside of it, and that is with some angling to get them in. I plan to look at the hampers Rhonda suggests.
Be very careful handling the dies. The blades are sharp! I've cut my finger a couple of times just picking them up somewhat carelessly.