Prepare main box parts
Select some nice 2x4 cedar and mill all the frame components. The frame rails use small stub tenons to join to the grooves in the legs, and now's the time to create spaces for them. Using a dado blade in your tablesaw, cut a 3/8"-deep x 5/8"-wide groove, 1/4" in from the edges on two opposing inside faces of all legs. Prepare the two lower front/back rails and lower side rails the same way.
The upper front, back and side rails are similar but less complicated. They only need one groove each. With the grooves cut in the edges of the legs and rails, you'll create those short, 1/4"-long tenons on the ends of the rails next. I used an auxiliary fence on my tablesaw that let me cut tenons with a dado blade.
If you want to follow the same process I did, set up your machine so the blade cuts a 1/4"-deep x 3/8"-wide swath, then use a mitre gauge to support the rails as they travel over the blade. This first pass creates the narrower, outside cheek.
Cut the ends of all your rails like this, plus some pieces of scrap that are the same size. These will let you tweak the machine settings to get the tenon thickness just right before proceeding with actual workpieces. Raise the saw blade to 5/8" high and continue the milling process to cut the opposing tenon cheek on one of your test pieces. See how well this fits into the panel board grooves you cut earlier and make adjustments until the fit is snug but not tight.
Repeat the process for the rest of your rails, testing them when you're done. If the fit is too tight, the side of the grooves may break off the legs during assembly. Clearly label all the parts top/bottom and left/right, then set them aside.
Choose the clearest lumber you have and cut the cedar panel boards 5 1/16"-wide x 23 3/4"-long before milling 1/4"-wide x 5/16"-deep rabbet grooves on the edges. Since these form the lap joints that connect neighbouring panel boards, they need to alternate in location. Take a look at the plans for details.
I set up my tablesaw to cut the rabbet grooves and rip the panels to width in a single operation. I stacked a right and left-hand pair of 1/8"-wide by 8"-dia. dado blades on a 10"-dia. rip blade with a 0.010"-thick spacer between them. Setting the fence to rip 5 1/16" wide also creates one nice 1/4" rabbet at the same time. Flip and turn the board around, making a second pass to cut the opposing rabbet. After this is the best time to sand the panel boards in preparation for assembly.
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