Veneer Tape? What? and, uh, WHY?
When you need simple custom built-ins: shelves, counters, cupboards, whatever, unless you have your own table saw and all that, you'll probably need to have someone else do the work for you. Or maybe not: if you get together a cut order (double-check it) and fax or email it to the right lumber yard, in a day or so a truck will pull up to your house and out will tumble a bunch of plywood cut more or less exactly as you asked for it, depending on which lumber yard did the job (if they use a sliding panel table saw, your chances are much better than if they use a vertical saw).
Which is all very cool, but the edges of the plywood will be rough, and show all the layers of veneer, and no matter how much you sand it, when paint hits it the ply's will expand differently and the layers will show. It's a look I guess, but if you want a nice flat smooth edge then keep reading.
You can get something called veneer edge tape, that you apply with an iron. For 3/4" plywood (if you can afford it, get "furniture grade birch"), get tape that's slightly oversize, like 25/32 or 13/16" or something, so once it's trimmed you'll have a perfect fit. Done properly, you can either paint over it and it'll look like a solid wood board, or once you get good you can even varnish and stain it. Note that birch tape has a finer, less-porous grain than oak.
• half-round file (one side is flat, the other side is arched, with a fairly sharp edge)
• small block of wood, around 2 x 4"
• sandpaper, about 120-150 grit
• iron, preferably one you can dedicate to the task so you don't get residue on your clothes
• stiff brush
The whole process will go much easier if you can figure a way to clamp your board to a work surface. As you'll want your work edge parallel to the floor, usually the easiest way is to screw two pieces of plywood at a perpendicular to each other; clamp one to the table, and then clamp the work piece to the other piece, with the result that your workpiece is perpendicular to the table surface. If needed, use little shims to protect the surface of the wood from the clamps.
First, use the brush to sweep all loose sawdust from the edge. If you neglect this step, the glue bond may be weak and later separate. Then get a piece of tape slightly longer than you need. Watch out for splices; you may choose to exclude them, or not. Save too-short pieces; you'll find a use for them someday. Then, start at one end, and begin slowly moving the iron (preheated, use a high setting) from one end to the other, pressing down. For long pieces, you may need to go back over it a few times to be sure it all stays warm.
Then, use your block, held at about 45˚ so the end is partway up on edge, to vigorously burnish the tape down, plenty of pressure. There should be just a little tape protruding over both edges, and no plywood showing at the margins. You need to check it now: lightly tweak at the edges of the tape, in an upward stroking motion, with thumb and first finger. You'll quickly learn to feel the difference between well-bonded tape (high tight sound and feel) and tape which needs more work (it gives a lot more and the sound is lower and softer). Go back over it with the iron and block until all of it is perfect. Pay special attention to the ends. Any tape that gets touched by the iron needs to be re-burnished, as heating it can loosen the glue.
Now, the tricky part. It takes a bit of practice (start on scrap wood), but if you try, eventually you can get it; once you do it's a breeze. Starting at one end, hold the flat side of the file against the tape, with pressure against the leading sharp edge, keeping the flat face slightly off the wood. The goal is to use this sharp edge (NOT the flat face) to *cut* a curl of excess tape, and not to use the flat side to file it off. Work in downward strokes, and try to stay off the plywood surface. When you begin to get it, you'll start to get thin spiralling curls of wood. Maybe in the beginning just go ahead and file it off with the flat side, but gradually try to get up on the edge; it's way more efficient, and doesn't leave as many file marks in the wood. To trim the ends, hold the file flat on the tape and rock it over the end corner; the tape should snap off pretty neatly.
Then give it a pass with the sandpaper. Hit the edges with a nice even 45˚ pass, then go down the edge with your fingers and feel for bits of glue or other roughness. Long strokes will give a more consistent edge; short strokes can easily take off too much material and leave you with an uneven edge. Look for any chatter marks from the file on the main faces of the plywood, and remove them.
At first this will seem really clumsy, and you'll waste some wood and time, but with a little practice you can get pretty fast at it. That and a can of paint and you can make pretty much all the utilitarian furniture you need. Currently I have a work table, a DJ stand, five shelf units, loads of adjustable shelves, a free-standing closet, a large bed with storage under and shelves behind, a kitchen counter, several closet doors, a desk, speaker stands...it just goes on.
Hopefully the diagram helps. Showing you would be easy; trying to make a free-standing block of type to accomplish it is a little different animal. Please feel free to drop me a line if you have any questions.
For assembly, your work will go much faster if you get a decent plate joiner (DeWalt is a good brand, probably around $200, and it'll last forever). It makes a cut in each piece so a flattened-football-shaped "biscuit" can be glued in. It takes a bit to place the cuts just right, but then assembly is a cinch, fast and accurate, way better than trying to hold two pieces in alignment while driving screws into them, a total pain.
This is a seemingly-obscure little skill, but when you need a custom-size piece, having this in your box makes it affordable and easy. New apartment? Carefully work out that cut list, send it off, and in not so long at all you have exactly the pieces you need. Don't forget nylon feet as needed so your piece will slide easily.