This plan should result in a fully collapsible beach chair that's actually surprisingly comfortable.
I didn't plan to make an instructable of this so the photos of the progress are missing but I've covered all the construction steps.
Step 1: Materials
All you'll need are two broomsticks of different widths and a beach towel - string, woodglue and/or hose clips are also useful but may not be necessary. Dowels work just as well but they're more expensive and the DIY shop near me doesn't carry a wide range, but broomsticks were easy to get.
I put walking stick ferrules on the sticks as well but that's not essential, just useful if you're going to use it on a surface other than sand.
Step 2: Building the Frame
Cut two 19" lengths off the thinner stick (your size may vary but I found 19" was the perfect height for me to lounge back against) and cut the thicker stick to three inches wider than the towel.
Drill two holes into the thick stick, leaving one inch of solid wood at each end. My first attempt only left about half an inch and the wood split with use, which I think was due to lack of strength. The holes should be just large enough to let the thin sticks slot inside although you may need to sand them down to ensure a good fit.
Either bind the ends of the thicker pole with string then cover with glue or use hose clips. Either way, since that was the part that split on my first attempt, I'd recommend strengthening it if possible.
Fit the ferrules on next. You may need glue or they may be tight enough on their own. I'd recommend not fixing them on until you're sure that the length of the legs is right for you.
Step 3: Sewing the Seat
Fold over one end of the towel and sew it into a tube large enough to take the thick pole. Make sure you stitch it firmly enough as this will be supporting your weight and you don't want to fall over backwards.
Fix two loops of elastic to the other end, although it might be best to leave this until the rest is done so that you're sure of the size of loop. Again, speaking from experience here...
Step 4: Assembling the Frame
Slide the thick pole through the tube on the towel (I've left it out to make it obvious, although I doubt it really needs it) before fixing the legs in. They should be tight enough not to fall out when you lift the back up.
Step 5: Try It Out
I found the best place to sit was just further down than the feet, that spot let me rest my shoulders just beneath the back. Lift the back up, letting the legs point behind you slightly and lean back. Adjust as feels secure for you.
It's hard to try to get a photo of this so I've used cushions instead.
Step 6: Pack It Away
Just detach the legs, place them inside the towel by the back and roll it up. Fix the loops over the ends and you're ready. In my case, I've got a 33" rolled towel that can be carried to the beach easily, used as a chair and then disassembled and used as a towel as well if I need.