Answer
Weight in orbit = $720.1\mathrm{N}$
Weight on earth = $733.7\mathrm{N}$
The man feels weightless in orbit since he is in a non-inertial frame and the centrifugal force cancels the graviational force.
Work Step by Step
Mass of astronaut $m = 75\mathrm{kg}$
Distance above the earth's surface $d = 600\mathrm{km} = 6\times 10^{5}\mathrm{m}$
Radius of the earth $R = 6.38\times10^6 \mathrm{m}$.
Mass of the earth $M = 5.97\times10^{24}\mathrm{kg}$
The gravitational force on the astronaut in orbit is
$F = \frac{GMm}{(R+d)^2} = \frac{6.67\times10^{-11}\times5.97\times10^{24}\times 75}{(6.38\times10^6+0.6\times10^6)^2} \mathrm{N}= 720.1\mathrm{N}$
The gravitation force of the same astronaut on the earth's surface is
$F = \frac{GMm}{R^2} = \frac{6.67\times10^{-11}\times5.97\times10^{24}\times 75}{(6.38\times10^6)^2} \mathrm{N}= 733.7\mathrm{N}$
Thus we see that there is not a huge difference between the gravitational force on the man on the earth and in orbit. The reason that the astronaut feels weightless in orbit is that when he is in orbit, he is in a non-inertial frame of reference which is rotating around the earth. The necessary centripetal force for this orbit is provided by the gravitational force of the earth on the astronaut. But since he is in a non-inertial frame, he experiences a pseudo force in the form of centrifugal force in the direction opposite to gravity but having the same magnitude. Thus, the net force the man experiences is zero and thus he feels weightless.
