I joined them with rabbet joints. The bottom is 6 mm plywood.
They are glued and nailed. I use these squares red squares I made out of oak and box joined to keep things square while I nail things.
My shop is so small everything is on wheels and most things have a double function. That bench is usually against the wall and I use it for finishing pieces, but some other times I roll it behind the table saw and use it as an outfeed table which is very handy to rip through big sheets.
We made this jig to figure out where each drawer slide should go and make sure both sides match up perfectly.
My father made the calculations and the sketch for that part. You can kinda tell he did that for a living for decades.
The drawer faces are made of 3/4 inch walnut plywood, the same used for the cabinet itself. The edges are made of solid walnut cut in strips of 6mm and miter joined.
I made this small jig to make the holes for the drawer pulls so they are in the exact same spot for all the drawers. All you need to do is draw a line at half the face and then line it up with the line on the jig, then drill.
The drawers have a 2 mm clearance (approximately). It took a ton of work, tweaking, sanding, wedging and hand planing to get a perfect fit.
I sanded it all with an orbital sander and then with 180 and 400 grit paper.
To finish it, I blew compressed air on all the pieces to get rid of most of the dust, then I rubbed it with mineral spirits to get rid of the rest, let that dry and apply a coat of tried and true varnish.
Turned out really nice.