Giant Nano-Material Sling Shots and Egg-Like Capsules for launches into Space would it work?

From the time I was six or seven I have always been fascinated with sling shots. The first time I saw my friend peg another kid with a rock slung from one I was hooked. As time progressed so did my fascination with sling shots. During my high school days we use to use the 3 man sling shot to launch eggs hundreds of yards.

My question is could we build a sling shot large enough to launch small space capsules into low orbit. If the sling shot is not strong enough or too forceful on it's occupants could it be scaled down so that it would be the initial stage of a space shuttle.

Hypothesis: I propose that a giant sling shot might be an extremely efficient simple machine for getting cargo(not people) into space.
Unless there is a way to make a capsule that can somehow reduce the g's drastically on the occupants there will be no way for the human body to cope with the ride.
However, if we end up finding that water and other cheap basic human resources like that do not exist in abundance on the moon or mars this might end up being the cheapest and most efficient way of getting those resources outside our atmosphere.

Where am I going wrong?