Black_Snow,
Lionheart is right, slings don't have to use any leather at all. The first eight slings I made were entirely of nylon twine (mason's line, it's called over here, it's used in construction to mark the level for the next course of bricks). My first sling, which I still use, was six strands of mason's line, braided just exactly as
Dan Bollinger
describes (this is the same article that Lionheart linked to). It cost me about US$0.50 and five hours of time.
Twine made of sisal or jute or hemp all work very well also. I found cotton twine to be too stretchy, but that could be my braiding rather than the twine itself.
By the way, if you are new to slinging, tennis balls make very good sling ammunition. They are heavy enough you can feel them in the sling, and they won't break your head when you shoot one straight up ("Now where did that stone go? ... THUD"). Or when you goof a release and slap yourself in the side of your head (that one hurt some, but not as much as if it had been a rock!)