A conex box is a good bit larger than most of the root cellars I have been in. Ours is a small room that we enter from inside our basement. The walls are poured concrete, and it has two 4" vents. It is located under our side porch. The floor is crushed stone, and we have a few plastic pallets in there to keep the vegetable off of the ground. Ours is about 8' x 12'. We don't keep that much in it, potatoes, pumpkins, squash, onions. We tend to can or freeze most of our home grown produce. If we have a lot of potatoes, we may put a gaylord bin in there to dump them in on one of the pallets.
A few things to remember about conex boxes, they are designed to support their weight from their corners, they are not strong on the top or on the sides. They also need to be and stay level or the doors will bind. If you do reinforce one and bury it, if one corner settles, you will have issues a lot harder to fix if you can not lift and re level it easily once it is buried. You will always have a step up into them unless you cut off the bottom door latches. The doors on a conex box swing out. Depending on your winters, if you get a lot of ice or snow, you may want a door to swing in instead of out or you may need to build a porch over the door. One additional issue you will have is rust from all the humidity. Root cellars are damp. Produce has moisture in it, and with only the small vents on a conex (unless you add additional ventilation, and even with the added ventilation), you likely will have rust & flaking paint issues.
Conex boxes have their uses, but I think there are better ways for you to plan a root cellar. Poured concrete walls or concrete block with a poured concrete roof would probably be most cost effective, for a long term root cellar.