This is a fun technique that was brought up in one of my favorite books,
Weaving on Cardboard. I'm in the dotee swap, and when I came across it, I thought, !, this is what I have to do! Because I'm currently obsessed with weaving... so, obviously.

end products. They're about 3-4" high.
So, you take a toilet paper tube, and starting about 1 inch from the top, punch a lot of holes (around 30) fairly evenly around the tube (for this, you can use a yarn needle). Then you use fairly sturdy thread and sew in and out twice around until there is a stitch between every hole on both sides (though, we only use the outside holes). Then I went around the tube with a pencil and put a number above every stitch. This number needs to be even, if you have an odd number, you'll just have to skip one, or double thread one.
Then you put the warp on:

for this doll, I was using sock yarn, but I would suggest something with a lot less stretch - cotton, or silk, or even single ply wool - the spring will get irritating later on. Put the needle and warp yarn down through stitch 1, and bring it to the other side, and go up the opposite one, if you have 32 stitches, this would be #17, if you have 28, it would be #15 (and I suppose if you had 30, it would be #16). Then move over to the next number over, and go back down and across the bottom to #2, and continue in this manner until you're done. If you're lucky, they will sit next to each other, and you can just tie a knot, if not, secure them as best you can.

add weft - I used a combination of yarn and roving for this one. just keep packing it in, pushing it towards the top.
*** the most important bit: you need an even number of warp threads to be able to secure the warp at the top, but you'll need an odd number of warp threads to weave in the round. So, push two threads together, and treat them as one for the rest of the project ***
you could use a crochet hook here, depending on what feels better. a needle is a bit faster, but for roving, the crochet hook is gentler.

now it's time to think about the bottom.

] start as close to the spot where the warp threads cross, and work outwards.
you want to keep pulling this tight, tight, tight. And pack on as much yarn as you can. The more yarn you get on, the more stable it is. I wouldn't worry if you can't match up the weft coming one way with that going the other way. At least, I didn't.


and when you're done, you just use a seam ripper to take off the thread that held the warp to the tube (Don't cut the warp!) and it's done! Stuff, sew up bottoms, embellish.
so, what's next? What about... oatmeal containers?

The principle is the same. use a nail, this time, punch holes... thread with string, warp as before. Again, a non-springy warp will save you some hassle. (although, I used one)
I would also suggest using a coarse sandpaper on the container, because as is, it's so slick, the warp tends to wander off.
Anyway, in this case, you have to start at the bottom and work up, just to stabilize the warp.

and if it's not centered, you have to correct for that. I did back and forth rows to fill up the empty side in a lighter color so you can see them. Keep pulling tight, keep pushing in as much yarn as you can
once you go up on the sides, more packing, more pulling, when you're at the end of a piece of yarn, then you over lap the next with it for 1-2 inches, and then leave the tail of the old thread and just go on with the new. I push the yarn down between the warps constantly. And when you're done, and absolutely cannot put on any more yarn, be sure the warp ends are knotted, and trim all the loose threads, the tension should hold them in.

use seam ripper, remove from container.

fold under ratty warp edge.
then line, add handle, and viola! A bag! (Alright, I haven't gotten this far yet.) But I think it's awfully cute as is.