I almost bought a bag that looked almost exactly like this for an 18th century period costume. The guy told me what it was called, but I can't remember. It was something along the lines of "sailor's sack" but that isn't it. Sailors used sacks similar to this for hundreds of years. They're very simple and made from heavy linen or canvas; heavy muslin also works. I checked a bunch of reenactment supply sites and tried a number of search terms but had no luck (which is really kinda bugging me >_<).
Fortunately, they're pretty easy to sew up. In fact, my first sewing project in 7th grade home economics was pretty similar to this. You sew a piece of fabric into a cylinder. You sew a circle of fabric onto the base. You sew up a shorter cylinder of fabric as reinforcement, and sew a 2nd circle onto that, optionally rolling a cord into the seam so it holds it's shape tighter. You turn both pieces inside out so the seams are all hiding. you fold the top edge of the reinforcing piece inward and top-stitch it onto the sack so that it looks like the picture. You can optionally tuck cord into that fold too. You then either fold the top of the sack inwards and hem it up so it looks like your picture, or more sensibly, you fold the rope into the hem before sewing it (being careful not to sew through the rope) and now you've got a proper drawstring.
For the tie-off string, use proper hemp rope if you can find it (not easy in the US), else use cotton rope. The rustic looking jute rope at the hardware store is incredibly uncomfortable to carry around.