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Simple butt joints, either nailed or screwed would be more than strong enough for what you are doing.
The problem would be an aesthetic one - the edges of the plywood exposed. If this is what you want to do, then automobile or woodworkers Bondo will smooth out the rough edges to where they can be painted. If you have the ability to miter the corners, that would solve the 'edge' problem. The miter joints will not be as strong butt joints, but they should still be able to get what you want. A vertical load using boxes should hold up the Empire State Building. (Figure of speech - don't want to get the engineers riled up)
If you are going to put liner boxes inside for storage purposes, I would make the inside of the boxes approx 1/2 larger than the liners. that way they wont jam when you tip the boxes upside down to get the liner out.
Have you thought about not even using a box inside the boxes? Plywood makes a good liner for clothing items.
Here is another idea about the boxes to resolve the 'edge' problem. You get or make 3/4" square sticks. Nail the sticks to the plywood edges and that will hide it. If you wanted to get creative, you could use 1" square sticks and the vertical edges can be left proud of the surface. If you are going to paint the boxes, maybe paint the vertical sticks a contrasting color.
Just remember to post photos of progress and the finished product
Good luck and have fun in whatever you choose.
Tony B Retired woodworker, among other things.
"Strive for excellence and settle for completion" Tony B
Last edited by Tony B; 06-10-2020 at 05:50 AM.