A small mechanic’s toolbox or a fishing tackle box usually works best. They’re sturdy and they give you enough compartments to keep cleaning supplies from turning into a mess.
I repurposed a lower Craftsman tool box, it has four drawers. I keep it in my garage and the bottom drawer holds my range cleaning box. The only thing I really don't like about it is it's on wheels and it rocks back and forth while running cleaning rods through rifles.
I paid $50 for the Craftsman box at an estate sale, but $155 at Lowes isn't a bad price either. The top is wide enough that my Tipton gun vice sits on top with room to spare. I also attached a piece of PVC pipe to it to hold my cleaning rods.