Like the other answers explain, mitered cuts is the way to do this. And using thicker plywood definitely makes all of this much more forgiving/easier. But even if you have real thin plywood, as long as you can "sneak up" on the right miter angle, it's not that hard to make this work.
Like the other answers mention, you need to get an accurate miter or it won't fit up right and the joint won't hold well. But it's not difficult to get "near-perfect miters" even on a real crappy saw with a miter setting (table, circular, radial, etc). You just need to sneak up to the correct angle. So, how do you do that?
Set your saw to what seems like 45 degrees. (You can use a carpenter's square, protractor, angle finder, etc to try to see if the saw blade is actually 45, but it doesn't matter). Cut two scraps of wood and fit them up on a flat table. Slap something square next to them and see if the pieces are really mating at 90 degrees. If they're not, adjust your miter, cut again, fit again, until it's pretty much dead on. Now you have a 45 degree miter.
If the angle still isn't perfect or the walls don't mate up perfectly, you do have two ways to "fix" the result.
If the angle was too small (<45) you'll have a gap when the pieces are supposed to be at 90 to each other. To fix this, smear some wood filler (sand something with super fine sandpaper, then mix that sawdust with wood glue and a little water) on the pieces before you clamp and glue the box together. Make sure a little extra filler is coming out the gap. Once it's glued up, sand down the faces & corners. You'll get a super sharp corner and a layman won't be able to tell there was a gap filled.
If the angle was too large (>45) then the edges will fit fine, but the inside of the joint will have a gap, so it won't hold together well. This has sort of a two-part fix.
2.1. Make the bottom/top piece fit the walls. (If it's not apparent from the other answers, you need to make a mitered cut on the top/bottom of each wall piece, and then cut a new top/bottom piece with all its edges cut with the same miter). Cut the top/bottom piece so that it's slightly too big to fit up with the walls. Then slowly sneak up on the width/length until it fits. The walls will mate up to the top/bottom even if they don't mate up with each other.
2.2. Use wood filler to fill the gaps on the inside where the walls connect. Since nobody's going to see this anyway, it shouldn't matter. If that doesn't hold, you can glue in the square piece of wood like in @Doresoom's answer. (I would cut the block into a triangle, so you can use one piece of wood to fit two corners; saves space/material).
If the plywood is thick enough, for a strong box, cut a dado into the bottom of the walls and slot & glue the bottom piece into the dado. Otherwise just edge-gluing a bottom piece into the inside bottom of the box won't be very strong. The top piece will always need to be mitered if you don't want any plywood banding to show.
I used all of these methods when making compound-angle trays out of only plywood and glue, because I suck both at math and at finding the right angle on my table saw. They came out pretty good, all things considered!