The solution: drawers in the cabinets, that occupy as much of the available area as possible, so if I want something out of the back of the cabinet I can just pull the drawer out and have at it.
I chose to make the drawers fairly low, riding on a single slide beneath the drawer. (I'd considered using two slides, one on each side, which makes the drawer easily removable (and much easier to install) but they cut into the drawer's width unacceptably.)
I used hemlock (because it's inexpensive and nice to work with) for the drawer sides. Poplar is vastly nicer wood for drawers, but these are going to be painted so I can tolerate less pretty wood that tends to chip, for less than half the price of poplar. These show the long sides and two fronts, for the upper and lower drawers. (The lower run has to be 1 cm narrower to compensate for the invisible euro-style hinge in the cabinet door; see below.) I set the tablesaw dado for 1/4" (the claimed thickness of plywood, about which more later) and ran the 8' hemlock lengthwise (low contrast pictures don't show the channel so well), then chopped it to length and ran a 5/8" dado across the front/rear sides of the drawer, to make a lapjoint and hide the bottom dado channel. (Otherwise, it would be visible on the sides of the drawer.)
Alas, 1/4" plywood isn't. It didn't fit. Luckily I have a router table, so I put a big fat bit in there and took about 0.5mm off all the way round the plywood. This will, of course, go on the bottom where it will be invisible. Now it's a nice press fit into those dado channels.
I sanded the sides and the bottom mildly (220g) so they'd paint smoothly and started assembling.
A pile of drawers press-fit together. There will be eight in all.
The first glue-up. I like pipe clamps, yessssss. If you ever get some, it's nice to make sure that the screw handles can rotate when the clamp is flat on the floor. Otherwise you have to put a piece of lumber beneath it, which isn't a problem, but is annoying.
There are thin backing strips that shim the inset bottom so the slide can attach along the full length of the drawer's bottom rather than just at the front and rear, cut from scrap 1/4" plywood.
Here's a slide, in place.
(Avast! Junk In Back Of Cabinet, Ahoy! I have a Tupperware problem.)
Note the protruding hinge, which necessesitates different upper and lower drawer widths. I planned the drawers so when their backs are against the back of the cabinet, the fronts would overlap the faceframe of the cabinet by 5 mm or so: not quite touching the back of the cabinet door but close. This precludes putting a drawer pull on the front, but prevents the drawer getting jammed against the back of the faceframe (the clearance is quite close.)
And a drawer experimentally placed (needs painted.)
(And a nice view of my poor taste in canned goods. Pineapple chunks are necessary for doing good sweet-and-sour stirfry.)
Since I'm an efficiency nut, I set this up as a sort of assembly-line thing. Ploughing the full-length dados just makes sense, and is safer. I did the plywood bottoms in stacks so they're all the same size, except for the first one (which I messed up in experimenting, and had to redo.) Likewise the cross-dadoes and the routed edges on the plywood. It took me about an hour to do the first drawer, and about an hour to do the next seven and redo the first one correctly.
Materials (per drawer) for 12" wide x 22.5" long: