You're looking for a line that is 2*sin(π/7) of the radius. That's 0.86777. The square of that is 0.7530, which is pretty darn close to 0.75 (1 - (1/2) ^ 2).
So make a triangle whose height is half the radius, hypoteneuse is the radius, and the other edge is 0.8660, within 0.001 of the real value and much more accurate than I can possible draw with a straight-edge and compass.
So it quickly turns into a question of perfect tools and other things that don't actually exist.
Somewhat pedantically, if it were archs and lines, I would consider it differently - they are hypothetical constructs and subject to hypothetical boundlessness.
But a straightedge and compass are not imaginary things; they are things of the material world and they are subject to material limitations.
Even one million is nowhere close to infinity, but the sum from x = one to one-million of 1/8^x is so stupidly close to 1/7 that you're most likely getting your toolkit delivered by the Archangel Gabriel himself.
And in less that ten minutes I could write that entire number to file.