I am going to be adding 2 lift out trays to the inside of the box. I am making cleats to hold them in place. This is pine I resawed and planed down to just under 1/4" thick

The longer one goes in first to support the top tray, then the bottom one over it to support the bottom tray. I applied some glue then tacked them down with some brass pins        

Now for the bottom. I want the inside to be a juxtaposition from the plain, painted outside. I will be using curly maple veneer. Since the bottom will not be painted on the outside, I am using 1/2" maple plywood. Veneer requires a stable and flat substrate. 

I start by cutting the veneer slightly oversize in all directions. The bottom is also slightly oversized. I will trim it to fit once the veneer is applied.      

I will be applying the veneer using an old traditional process called hammer veneering. The process involves using hot hide glue and a veneer "hammer" -  which is used as a squeegee to press the veneer to the substrate and squeeze out the excess glue.

The tape is not used to hold the joint together, but to mark the orientation. I usually use a cabinet makers triangles, but I've found hammering over pencil leaves black marks that don't come out

First I start preparing the glue. I use 192g strength hide glue granules. I hydrate the mix overnight then bring it up to the temperature of 140 degrees. The temperature and viscosity are critical. I use an old jar and a hot pot express for a glue pot

The next step is applying glue to the substrate. Since this is plywood and I know it is flat, I didn't tooth it. I apply it to the side I am veneering first

Next I apply the veneer face down in the glue, and apply glue to what will be the back of the veneer. This does a few things
1 - Warms the glue under the veneer on the substrate again
2 - Maintains an equilibrium moisture level in the veneer
3 - Glue will transfer to the face and provide lubrication for the hammering process. Unlike PVA glues (titebond etc), hide glue has no effect on finishes so saturating the piece in glue will not have any ill effects later on.

Now I flip the piece again and hammer it. I use the hammer to press the veneer to the substrate and squeeze out the excess glue, Making sure there are no spots lifting or not adhering. It takes some practice, but isn't too hard.

Now for the other side. Same process. Glue on the substrate, veneer face down, glue on the veneer, then flip. However I do not line up the edges here. I overlap them so the second sheet is on top of the first one by about 1/4"

I then use a stainless steel ruler and cut the joint. The glue on the veneer keeps the ruler down, and washes off easily in the sink later. I cut through both the top and bottom veneers. Before I peel up the waste after cutting, I add more glue to the joint area to re-heat the glue under it. I peel the waste from the top piece, use the x-acto knife to lift the top piece a bit, remove the waste from the bottom piece, then hammer the top back down. This leaves a good looking joint

Done, I scrape off the glue a little with the hammer and put it back in the glue pot to be used later. As mentioned, hide glue does not affect finishes so there is no reason to be diligent here

After about a day, I scrape the veneer smooth with a cabinet scraper. Then I cut the bottom to exact dimensions to fit on the box

I taped off areas that will receive glue and applied a few coats of dewaxed shellac

Now to install it. This procedure works no matter what bottom you choose. I set my marking gauge about half the thickness of the box, then mark all 4 sides of of the bottom (on the bottom side). This will give me a good reference to put my nails so they don't end up going through the edges

I mark on my scribed lines every few inches with an awl. This will be where the nails go

I apply hot hide glue to the bottom of the box being careful not to get any on the inside. Liquid hide glue will  work well here too

I press the bottom on the box and hold it in place for a few minutes so it won't slide around

I like to pre-drill for nail holes to avoid splitting. I put one in each corner to hold the bottom on, then do the rest. After all the nails are in and the bottom is attached, I move on to the lock