I have heard many anecdotes of shepherds using the sling to kill marauding foxes and coyotes. In each story, only one rock was slung at a time and the slinger had sufficient accuracy to kill, not merely frighten, the animal.
Ralph Ray Craig
When I was a kid, up until I was about 16 had a car and could go fishing on my own, I did a lot of messing around (a good way to characterize what I was doing) with throwing stuff (sticks, spears, knives, and slings) and made my first slings. I never thought of the heart to hand measuring system but that does come out at about 30-36 inches which is where I ended up. At that time I would buy leather shoelaces for the straps and use soft waste leather from my Dad's leather projects for the pouch. Have used it to harass a friend in a canoe on a lake so I know that you can heave a good size rock quite a long way. I never got very good for accuracy but I could heave a rock a pretty good distance, 60-70 yards. I used to think it was closer to 200 yards :). I used the thing to scare game towards me by throwing the rock to the other side of brush piles, cedar clumps, etc.
I've hunted small game with it, birds, rabbit and squirrel mostly. I've had limited success, but haven't been using it with the aim of using it to specifically hunt with. If I were to do that, then I imagine I'd be able to bag game OK, up to the level of a fox. Beyond that, I don't know. Of course, this depends on the range as well. However, I have sent a steel ball 1" diameter humming into a tree trunk some 80 yards away and had it hit with a very solid thud, so obviously the size of the ammo affects the range. (Sound effects courtesy of a very impressed group of people I was demonstrating to.)