Birch build would be easy.
For best strength, cut the bottom so it is trapped by the sides. Apply the long sides to the bottom. Then size the ends to overlap the bottom and sides. That puts all your fasteners so the weight / gravity works on them in a shear, instead of trying to pull them out of their holes.
Glue and brad nail the sides, then go back and countersink and screw all the joints. Wipe the excess glue out of the inside joints. To reinforce it you can glue in strips of material inside on the joints. That will greatly strengthen the joints without taking a lot of space and allow you to make the box out of thinner plywood.
I did a similar box a while back as a transport case for my telescope, thin ply.
Sturdy enough and this way you can custom max-size your container.
eta given the size of the box you want, the stuff you want to put in it, you might consider building a chuck box, the sort that nestles inside it's own base stand.