My ironing (pressing) station is a converted baby changing table. There is a shorter part that has three drawers, now overflowing with layer cakes, jelly rolls and charm packs. The taller portion is the perfect place for those oversized dies. When I get that stinking phone set up, I'll post a pic.

I purchased two wardrobe cabinets off of two different Craigslist ads. I paid $20 for one and $25 for the other. They are Sauder brand and they seem to come up for sale often in my area. I make labels with a larger font so I can see them in the low light corner of the basement I store them.
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Thank you for the additional suggestions. My first shipment of dies arrived today. Those things are heavy! One of the boxes they arrived in was badly damaged on one end, but the dies appear to have survived okay. The cutter hasn't arrived yet. <sigh> I'm so eager to start using it!

I have my long dies on top of a shelf in my sewing room and two of the storage racks side by side so the long dies just slide into them. They are heavy and I figured they could being over time if standing the long way they may bow like any board will do. The other dies I have bought the storage boxes. I don't have the studio dies. But I do know my strip dies and train die are all heavy so I prefer to slide it.