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I did the math off the top of my head, so let me do this in more detail this time ... check me on this.

A = v^2 / r
if we solve for r we get
r = v^2 / A

or for G's conversion

r = v^2 / (G's * 9.81)
so plugging things in...

r = 8000 * 8000 / 78.4
r = 64000000/78.4 meters
r = 816327 meters
diameter = 1632653 meters or 1633 km (rounded)
I must have flubbed up on my decimal point here.

STILL, that is a 1,000 mile diameter circle. If you planted the center in eastern Kansas, the western boundary would be up in Colorado and the eastern boundary would be somewhere in Kentucky. Basically trying to fit it in between the Appalachians and the Rockies. North part would be up by the great lakes and the southern would run through Texas.

Now, I'm sure that with some engineering, 13Gs for a satellite would be doable. That 500km radius MIGHT be doable. A 25G satellite would be even better but I don't know if we currently have the technology to build electrical components esp computer chips and solar panels to sustain that level of G's.

Now if you could combine this with a recovery vehicle that would carry the satellite but provide lower atmosphere air breathing thrust to offset drag, and perhaps a small booster rocket for high altitude, then you could get away with say 5 km/s.

So recovery sled, plus booster, plus 25Gs would be a radius of:

r = 5000 * 5000 / (25 * 9.81)
r = 25000000 / 245.25
r = 101,937 m
so a 205 km or 127 mile diameter ring. Now that seems more realistic to me. That could be done inside of Texas.

It would be interesting to compare that to the length of a rail gun required to get the same configuration up to the same speed. A cost comparison would be interesting as well. Esp since the ring would need to be build in such a manner as to withstand potentially dozens, maybe even hundreds of tons of lateral force.