Decide how big you want your box (L, W, H) with pleasing proportions, rectangular or square. Cut (rip) a piece of plywood along the grain that is at least 2L+2W long plus a couple of inches longer for cuts, and bit wider than H. Roughly measure W, L, W, L along the length of your piece and mark these lengths 1, 2, 3, 4. Find another piece that is a bit larger than LxW for the top. Mark this piece along the edges 1,2,3,4 with 1 and 3 being the W edges and 2 and 4 being the L edges. You will also need a thin piece of plywood (~¼") for the bottom of the box that is slightly smaller than LxW.
Set your saw blade at 45°+ (maybe 45½° or 46° if you can do that -- this will make assembly lots easier later on). Set the fence at H and run your long piece through lengthwise, with the nice face up to get a 45° bevel on what will be the top edge of the sides.
Set fence at W and run your top through along the L edges 2 and 4, nice face up. This will give you the top W wide with a 45° bevel along both the L edges.
Move fence out of the way and set your miter gauge at 0°. Run one edge of the top (1 or 3 W edge) through with your miter gauge to get another 45° bevel so that 3 sides (L, L, W) of the top have the 45° bevel. Leave the other edge square for now.
With your blade still at 45° cut your 2L+2W piece in half crosswise using your miter gauge, nice face up. These pieces should be a bit longer than L+W if you cut the whole piece longer than 2L+2W. Cut the other 3 ends the same way so each end has a 45° bevel. 3 edges of each piece will now have a 45° bevel, and the pieces will be marked 1,2 and 3,4.
Attach a shim block on your fence using a scrap of your material as offset. Put another longer piece up by the blade (which is still at 45°). Place your top side 2 or 4 along the blade and move the fence with the longer shim piece until it just touches the other edge (4 or 2). The fence should be set properly now to cut a side at length W. Take the 1,2 or 3,4 pieces and place the 1 or 3 end against your shim block. Now make a crosswise bevel cut by pushing it along past the shim block with your miter gauge. This piece will now be the same length as your top W edge, with 3 bevels for top edge and corners. Now you have the two ends of the box (pieces 1 and 3) exactly the same length as your top is wide!
Set fence/shim at L on the tape and run the remaining edge of the top through with the miter gauge to get the top with 4 beveled edges, LxW.
Using the miter gauge and shim stop, run pieces 2 and 4 through to get the remaining corner bevel. These pieces will be exactly L long.
Test assemble your pieces now to make sure everything lines up. Use some masking tape to hold the sides together while the flat bottoms of the sides sit on the bench. Set the top on gently and it will square up, and all edges should fit tightly. This is where the bevel cuts at 45°+ help get the edges tight, especially if you are using a thin veneer plywood. When the box is glued up the small gap on the inside of the joint will fill up and will not be noticeable.
Remove the shim, set the blade at 0° and half the depth of your material. Set the fence at about ½", and run the flat bottom edge of the side pieces through to get a dado for the box bottom to fit in. You will need to adjust the fence and run these pieces through again to get it the right width for your bottom to fit. Two passes will do it for a ¼" bottom. Or, you can use a router table with a ¼" bit to cut the dado.
Next step is to cut biscuit slots in all the beveled edges of the sides and top. How many slots to cut in each edge depends on the dimensions of your box. You need at least one, and two work better. If the dimension is more than 6" you can get 2 #0 biscuits in each edge. Just be sure you don’t cut one that will interfere with the bottom fitting in its slots. You might also want to keep them away from where you will cut the lid off in step 14. Cut these biscuit slots evenly along the edge, with the slot starting near the inside so it does not cut through the face of the piece. For example on a 9" edge cut two biscuit slots on centers 3" from the ends of the edge. Marking the centers on each adjoining edge is done most easily with the box taped together. Set the depth on the biscuit cutter fence to about ¾" for ¾" plywood and use the outside face as your reference for the biscuit cutter fence. For ½" plywood you will need to adjust accordingly to make sure the slot does not cut through the face of the plywood. You will have to hold the biscuit cutter with the sharp edge of the work piece in the fence corner, and the cutting blade against the beveled edge. This actually works easily but just hold it carefully.
Assembly of the box goes quickly, and here is where this method really pays off. Put a little glue in each biscuit slot, and insert biscuits tightly in the slots on both edges of opposite sides (1and 3 or 2 and 4). Fit biscuits tightly in all the slots of the top. Smear a thin layer of glue along all the edges. Set the sides on their flat bottoms, in sequence 1-4 so the grain pattern runs around the box. Align the bottom piece in its dados [NOTE: put the nice face of the bottom piece DOWN so it will be visible -- the face in the box will get covered with felt later], and get the lower biscuits started in their slots a bit. Line up the top biscuits so they just start to go in the slots. Leave the top edges angled out slightly so that the top biscuits can fit in the slots on the top edges of the sides. Now take the top and drop it down so all the biscuits line up with the slots in the top edges of the sides. Squeeze the whole thing together so that the biscuits pull all the pieces together. Pushing down on the top will tighten up all the pieces and square up the box. Check square and edge alignment. If it is bit off, or all the edges are not tightly fitted, give the top or sides near the edges a whack with a rubber mallet or hammer/block. [Having cut all these bevels at 45°+ gives you a bit of play here to get things lined up properly!] Have 6 clamps ready to clamp it all together -- two for each side and top, oriented such that all the edges are held tight, and the whole thing is square on all edges. Let it sit until the glue sets up well then remove the clamps.
At this point you have a closed box that can be easily sanded to get a nice smooth finish. Finish it as you would like it with paint, stain, wax, oil, varnish, or whatever.
To cut the lid off the box set your table saw blade [at 90°] to the same depth as the thickness of your side material, and set the fence at whatever distance you want for the lid thickness. 1.5" works well for a box that uses ¾" material. Run each side through the saw carefully, with the bottom toward the fence. Be careful not to let the top or bottom twist on the last cut when the pieces come apart. Also be careful not to scratch your nice finish. Note that this kind of cut will give you flat matches on the cut, and you will see the veneer layers of the plywood. [Some people have remarked that they like this look.] You might need to run some extra fine sandpaper very gently along the cut edges to clean them up a bit.
You can do the lid two other ways that require a bit more work in step 10. If you want the top to fit inside the bottom with some overlap, you will need to cut dados on both the inside and outside faces of the side pieces, each a bit deeper than half the thickness of your material, about ¼" wide, parallel to the top edge. The inside/outside dados should not be too close to each other as the pieces will split apart before or during assembly, and that will make getting the whole thing together properly more difficult. Leave at least ¼" - ½" or more between them. [Note that you need to adjust your box dimension H higher by this separation amount plus the width of the dado so that when you cut the box top off you have the box H high.] If you do not want to see veneer you can cut a wider dado in the inside of the side pieces, parallel to the top edge, centered on the depth of the box top, just slightly less deep than the material is thick. Let in a piece of wood that will fit the dado and lay flush with the inside surface of the box. When you cut the top off, by either method, you will have a clean look with no veneer as the cuts will be through this solid piece of wood.
Hinges (if you want them) -- Keep the box lid oriented properly to the bottom so the grain stays matched, and lay the box down on its front face. Position your hinges where you want them with the joint straddling the cut. Use a punch to poke some small holes for the screws. I find that with veneer plywood you can punch through the top layer and a bit into the next, softer layer. Put a bit of candle wax on your screws and screw them in easily. You can also put a small latch on the front if you want, but many people find the clean uncluttered look more appealing. If you choose to use hidden hinges you will need to cut them into the edges of the bottom and lid. Make sure that the sides are all still aligned tightly and there are no gaps, overlaps, or twisting. Adjust the hinges as necessary to eliminate.
Final finishing of the inside can be done using sheets of stiff felt (they come in 12x18 sheets in many colors) that are available at hobby and crafts stores for less than a dollar a sheet. These can be cut to fit each inside surface. Use a steel rule or square to get the cuts with a razor knife after making little nicks at the proper dimensions when you hold a piece of felt to the panel it will fit. Cut all the pieces and do a test fit. Use spray adhesive (available in the craft/hobby shop, probably near the felt rack) to spray the inside of the box as well as the back of each piece. It is best to start with the large pieces that fit the bottom and the top, then do the sides so they lay over the bottom/top at the edges. A clean edge between the box and lid edges can be had by using a scrap piece of wood against the side at the edge and running a razor along the edge. You could also add other little shelves, dividers, etc. for jewelry or other items. Lining with cedar makes a nice box too, especially as a cigar humidor. For larger boxes a little brass chain on each side or a catch will keep the lid from swinging back and breaking the hinges or pulling them out.
You might also want to put some finish on the edges where the lid cut was made, to match the finish on the outside of the box.
The lid can also have a recess cut into it for a needlepoint piece or inlay of some sort. This will add more time to the basic construction but will also give you more options for use.
This is a great project to do with kids. Depending on their age you might have to do the saw/biscuit work with them watching, but older ones can learn and do a lot with your help. Needless to say, get them goggles and ear protectors to go with yours. The measuring, sanding and finishing are something even a young child can do with help, using a palm or random-orbit sander as these are not too dangerous. The way the pieces go together, and the outstanding alignment, gives them a lot of satisfaction without a lot of twiddling. And having a beautiful treasure box made in a couple of sessions of their attention-span time is quite satisfying to both master and student. NOTE: This was tested on a 7 year old, who made this box for a cub scout badge. I cut the pieces, he did the gluing and assembly in one short session, sanding and finishing in another, and hinges and felt at the end after the finish was dry.
