I would recommend getting a shelf and bins. Over the years I have found several ready-made book shelves or garage storage shelves that will fit the plastic bins made by rubbermaid and Sterilite. The problem with that solution is that you need to use the lids which is a pain and not necessarily escape proof.
If you are handy at all and have access to some woodworking equipment you can make a custom rack very easily.
I made several 5-slot lidless racks using 3/4" plywood. These can be made for about $100 each including heat and bins. I buy the plywood in 24" widths so all I need to do is cut them into 6-18" segments for the top, shelves and bottom and 2-36" segments for the sides. The result is a rack that is 18" wide x 24" deep x 36" high. You can stain or paint them how you like. They're not hard to make and I have never had an escape.
John