Remember that every sheet of plywood should have two good long edges, two good short edges AND possibly four square corners1. Use this to your advantage if the factory edges are in good condition; and if they're not perhaps a little hand planing or sanding with a block will clean them up enough that you can use them as the bottom edge in drawer boxes.
don't think I have enough space to set up to feel safe using a circular saw (certainly no room for a table saw).
FWIW circular saws are considered the tool for making at least the initial break-down cuts on sheet goods, so if you will be using plywood etc. a fair bit one can be a worthwhile purchase. And arguably the safest way to use one to do this is to work on the floor, on top of a sacrificial surface such as thick insulation foam, so the method doesn't require a dedicated workplace.
Also, with just a few scraps of wood and a little ingenuity you can quite easily create a very capable sawing guide, saving 'a little' over some of the commercial track saws ^_^
Popular opinion seems to be that you can't cut straight with a (handheld) jigsaw.
Well popular opinion is wrong in this case. As I say in my Comment above, the (many) people who say this are either blindly repeating a 'fact' they've heard, or they're relating their own experience where they did it wrong.
Sure, with a jigsaw you won't easily match the kind of cut you can achieve with a table saw, or a circular saw for that matter (even used freehand), but you can cut at least as straight as with a panel saw and nobody goes around criticising hand saws for how bad those are at sawing to the line!
Just as when making cuts with a hand saw, so much comes down to the user.
With a jigsaw if you push the cut you're setting yourself up for failure. But slow down and be patient, let the saw do the work, and you can saw to line really well.
Does the blade really wobble side to side that much?
AFAIK it's not really about blade wobble, it's more about blade deflection (from pushing the cut) which can tend to pull the saw away from the desired path.
Slow right down however and a wandering jigsaw will start to behave itself. Sawing is not a race.
And that's just if sawing freehand. Things get even better if you jig it.....
Can plywood be (hand) planed, or does the glue gum up too much?
Obviously plywood quality varies (lots) but yes you can definitely plane it.
You'll probably find you need to re-hone your iron a little more frequently than normal, but not excessively so. I recently had to plane quite a bit of OSB which is way tougher than a typical plywood, and even that wasn't too bad2.
Watch out for spelching at the corners due to every second ply being at right angles to the edge, and utilise one of the end-grain planing tricks for clean corners.
What plane to use
Honestly just about anything. If the drawers you plan to make are not small you could do with using a no. 4 or 5 for their longer soles, but you can do this quite well with a block plane. Just needs a little more care to ensure your edges end up straight if you're using a shorter plane.
Regardless of the plane you use having the iron nice and sharp will of course pay dividends.
Sawing plywood by hand
I don't want to dissuade you from trying to saw sheet goods by hand, because with one of the better modern panel saws with tough, hardened teeth and the right setup (saw horses etc.) you can break down plywood surprisingly quickly3 and easily by hand.
Your first few cuts are unlikely to be the straightest ever made naturally, but as you're already committed to cleaning up the sawn edges by planing as long as you stick to the waste side of the line it sorta doesn't matter :-) And you will get better at it with practice.
So don't dismiss it as unworkable method, despite the undeniable advantages of doing it with a power saw. Sometimes it's just way faster to do it by hand, and good enough.
1 At least two have a reputation for being reliably square, but never take this for granted. Always check.
2 And this was with stock irons in very much non-premium planes, so not using A2, PMV-11 or anything like that :-)
3 3/4" material saws at approx. one inch per stroke according to Jim Tolpin.