Cut out the top, bottom, back and sides for the box out of plywood. The drawers will be built later.
Mark each side of the box to cut dadoes for the drawers to slide into. Spacing is critical in this step. The dadoes need to be cut in the same place on each of the two sides. Leave enough space between the dadoes so that your tallest router bit fits into the drawer. Also, allow some room beneath the bottom drawer. The dadoes will not be cut all the way across the board. Make the last mark 1/4" from the back of each side to show where to stop the cut.