Need Numbers to Complete Crazy Idea Proposal...

The numbers tell you in an eyeblink how crazy this really is. That's the value of being able to do back-of-the-envelope calculations -- you can dismiss dumb notions quickly.
Orbital velocity is about 16,000 mph, or about Mach 21. You need that speed to get into orbit no matter how high the launch point is. That is, if you managed to get the astronaut all the way to orbital altitude in the balloon, you'd *still* have to fire him at 16,000 mph. Now, the human body can tolerate an acceleration of about 6 gees for a duration of several seconds. Any higher acceleration or longer duration and you damage the astronaut, which would not go over well on the evening news. Getting to 16,000 mph (23,400 ft/s) at 6 gees takes about 120 seconds or a full 2 minutes of sling-shot slinging. Now you can calculate how long the bungee cord has to be to sling an astronaut from rest to 16,000 mph over a 2 minute sling. You may be amused to obtain the result that the stretch distance of that rubber band has to be about 1,400,000 feet, or 270 miles, which is about 3 times further than the orbital altitude you're trying to get to. So unless you want to dig a hole 260 miles deep to stretch back a rubber band far enough to launch an astronaut 90 miles in the sky and not turn him into a spacesuit full of blood sausage, I'd say you have a feasibility issue.
Compound this with the fact that helium is actually a pretty rare substance on the earth. You want a really really HUGE HUGE helium balloon, but you've got no idea (I guess) whether there's enough helium available to fill that HUGE HUGE helium balloon up.
Now, I hope that isn't being too much of a Negative Nellie.
PD

Agreed you probably would not want to do this with a manned ship. But let's suppose it was only for cargo. Hardened computer chips in some artillery shells can withstand accelerations of 10's of thousands of g's. At this acceleration the distance now would only be in the 10's of meters range. The question now is are there materials that could withstand this kind of stretching to induce such high velocities? This page gives some values of strength of materials and velocities that could be withstood by the materials:
STRUCTURAL MATERIALS.
The "characteristic velocity" listed in the table is the maximum speed the material could withstand if you wanted for example to make a flywheel of this material. You see to get up to orbital velocity in the range of 7000 to 8000 m/s only buckyball material, same as for carbon nanotubes, would suffice IF you wanted to do this by for example twirling the craft at high speed in a circle with a rope made of the material.(However, it might be you could get higher than the speeds listed by using tapering.) But it is not clear to me this "characteristic velocity" is the right speed to use for the idea of using a stretched elastic band. It might instead be the longitudinal sound speed in thin rods listed on the "STRUCTURAL MATERIALS" page. And this speed for several high strength, low weight materials already in common use is higher than orbital velocity. A related question could you use these high longitudinal sound speeds to induce propulsive mass to high exhaust velocities? A possibility: have a molecularly thin plate sticking up perpindicularly to the rod. Then as the molecules were vibrating at the high speed the plate would be made to move back and forth also at the very high speed. Then a gas used for the fuel could be made to move at high speed by being pushed by the plate. You would need to remove the gas from the plate once it reached the highest speed.
Bob Clark

Dear Robert Clark:
... What is the point? They will vaporize passing through the atmosphere. Perhaps you did not notice that even the tenuous atmosphere at 60 miles was enough to vaporize a space shuttle at less than full orbital speed. And you want to do this speed at ground level?
David A. Smith