I can't claim to have thought of this idea. I've seen many all over Pinterest. There are a lot of options for a diaper box. You can glue paper all over it, paint it, glue fabric on it, sew liners, etc. Really if you look you'll probably find a way to revamp it to suit your needs.
That's what I did with this. I don't possess sewing skills outside a basic blanket stitch, so sewing was out.
I don't have a lot of fabric on hand (reason: I can't sew), so even gluing it all over a box was out.
The only wrapping paper I have on hand right now is Christmas, so that was out.
I did have leftover paint from when my husband painted my workspace, scrap fabric from recovering chairs years and years (and years) ago, a glue gun, and an idea.
Start with a diaper (or other) box. Cut the flaps off the top of the box (and out of the handle if your box has them).
Start painting. I painted all over the outside (even the bottom) of the box.
Since the box is glossy, you really need to let the coats dry to cover as much of the print as you can. So add a coat, go read for a bit, then come back and add another.
Once the box is dry, measure your fabric against it. I made mine about 3 inches wide and the length of the box (2 shorter, 2 longer).
Glue a hem for the fabric (reason: I can't sew). I glued a hem by gluing a little bit, folding the fabric over, then continuing. If you can sew, sewing a hem would work too.
I wanted it to stand out a bit more, even after adding the fabric. I cut these freehand (really? can't tell!) and glued them onto the box.
Then you have some boxes re-purposed for your needs :) I plan to do at least one more for these shelves, or if my 2 year old's potty training goes the way it has been: for every shelf in the house. It's a long battle.