If you have curved corners, you will have some serious challenges and won't really be able to use butt joints. If the box is square, you can use your router to make a kind of tongue and groove arrangement Ir will be rabbits fitting into grooves, a simple drawer joint. This is a pretty strong joint, simple to make.
Glue a top and bottom on, or cut a groove for the bottom and a rabbit for the top, glue in a top, then cut open the top. You can use simple hinges on the top or so some semi visible barrel hinges, depending on how thick the sides/back/front is.
If you want it fancier and stronger, add splines.
You could also do it with mitered 45 degree corners and use a hidden spline for strength. This is a simple joint, but the 45s must be accurate. This is best done on a table saw to make the basic cuts, but could be done on a router with a 45 degree bit.
The more I do, the less I accomplish.
Last edited by DesertRatTom; 02-07-2020 at 11:40 AM.