As SaSSafraS1232 mentioned making a box for the purpose you stated is much like making a drawer.
1. Groove all sides, front and back of the box/drawer, then glue the sides to either the front or back piece than slip the bottom piece of plywood into the grooves, (no glue necessary) and glue on the other end. This captures the bottom completely.
2. If you wish to be able to someday replace the bottom, then make the back piece only as high as from the top of box to top of grooves so you can slide out the back for replacement. You can secure the bottom from slipping out with a couple of small nails.
General rules. A groove should be at least as deep as the bottom is thick and up to as deep as half the thickness of the sides. Allow 1/16" for expansion of the bottom panel during humidity increases in the environment. Make the groove as far from the bottom of the sides and ends as the bottom panel is thick (minimum). Preferably as far from the bottom as half the thickness of the sides and ends.
If you are using the box for the sewing contents, I would use a 1/4" panel, if it is for the sewing machine, I would use 3/8". For my shop drawers, expecting up to 100lbs., I use 1/2" plywood for bottom panels and 3/4" for sides and ends.
To learn proper box and drawer making, there are some very good tutorials on Youtube. If you want to build a box style, search "make a heavy duty box", if you want to build a usually more flexible and stronger drawers style, search "make heavy duty shop drawers".