S-4's idea will work, but I would take a different tact to it.
MDO does not have a paper
face much less paper layers. It's particle board that's a
bit more strong in using finer saw dust making it up. Are you thinking
MDO plywood that is a good grade of plywood with a paper
face on one or both sides? Good stuff, very strong. Or are you talking about some form of hardboard that uses less dense layers of paper or fibers and is available in thicker sheets?
In any case, the
MDO should work at least decently for your purpose when as with S-4 says, glue and the 2" or 2.1/2" screws to a
point. That
point is the stress you put on the joints when it's dropped. Given you will want to pre-drill and
countersink the holes. The 6" on center is decent but I might look towads 9" on center for the screws and 2" screws instead. Reason for this is the more screws you have and the deeper you get in after a
point, the more they are going to act as if a
wedge in tearing apart the layears of plywood when dropped or stressed. Given 3/8" thick plywood attached to the 3/4" plywood, a 2" #6 screw would be about the maximum I would use. Even a #8
wire size screw will be creating a larger
wedge even if a #6 is stretching it in sheer/
breaking strength. Perhaps a 8" on center to make up for it.
This is given plywood or
MDO faced plywood. If you are really using medium density fir or fiber pine, it holds up better than particle board to abuse but still is not what I would tend to want to put a screw next to the end of, much less would trust a 3/8" side panel to in the possiblity of there being something under it as dropped. Just might break
thru the
MDF and be a hastle to repair once glued.
MDF will have much more trouble in splitting between layers than plywood that has more fiber to absorb the screw. It also will show more dings than plywood when it's dropped and there is a possibility that the 3/8" thick stuff will crack or split around where the screws are once stressed, especially if too close to the
edge.
How about plywood that's sanded, laminated or sethat you install a paper or fabric
face on? A fabric covering or even fiberglass covering will add a lot of support to it.
The best engineering plan for any of the sheet
goods would be a sub frame inside the box to attach to. This way you are going into the side grain of lumber, not the end grain of sheet
goods, and your attachment screws will be further away from the
edge of the material. It will also give much more support to the box in providing a larger glueing surface for attachment. The sub frame will also support the thin materials sides as they
butt up to each other. Note also that the 3/8" plywood without support will not support a lot of weight or falling abuse without a frame. What's the length of the box anyway given the 3/4" thick sides are 8"x24"? Given this, 8"
thickness and no matter what the lumber type I might look to using 1.5/8" screws instead of even 2" screws.