[Lewis Thomson (OP)]

Werner is looking for answers to this question.

"How many kilometers from the earth's surface does weightlessness begin and does it depend on the weight or mass of an object when it starts to float? If we have a 100kg and a 50kg astronaut, which one starts to float first?"

Leave your answers in the comments below...

[Halc]

How many kilometers from the earth's surface does weightlessness begin and does it depend on the weight or mass of an object when it starts to float?

1 cm up is enough. Being weightless simply means the ground isn't pushing up on you at the moment. It is the proper acceleration exerted by the ground (or anything else such as being pressed into your seat by a rocket with the engines on) that defines weight.

[chiralSPO]

To clarify Halc's (correct) answer a little more:

Astronauts in the International Space Station (ISS) experience weightlessness, but it is not due to their altitude (at an altitude of ≈400 km above sea level, the acceleration of gravity is only about 10% less than at sea level; 7000 km from Earth's center vs 6600 km, and acceleration is inversely proportional to the distance squared). It is the earth's gravity that keeps the ISS in orbit, after all!

Those on the ISS experience weightlessness because the ISS, and everything/everyone on it, is in free fall. They are literally falling around the Earth! You can experience the weightlessness of free fall at any altitude, you just have to be falling freely. So from Halc's proposed 1 cm altitude, you could be in free fall for a small fraction of a second, before finding the ground (unless you're falling down a well). For more than a split second of free fall, one can skydive (I would never do this unless I had to escape a burning plane, and even then, I might pause...), or ride the "vomit comet" entertainingly demonstrated here:

[Origin]

"How many kilometers from the earth's surface does weightlessness begin and does it depend on the weight or mass of an object when it starts to float? If we have a 100kg and a 50kg astronaut, which one starts to float first?"

I think you are being confused between being weightless in orbit and being weightless because of no gravity.  When you are in orbit you are really in freefall.  That is to say you are in a gravitational field but you are in a continuously falling towards the earth.  It is like you are in an elevator with the cable cut so you and the elevator are simply falling.  In the falling elevator case you would be weightless so if you were holding a ball for instance and you released it the ball would float because it and you would be weightless.  In the case of an orbiting space ship the ship is falling toward the earth but the ship is moving very fast so it continually misses the earth. 
What that means is this; when you throw a ball it travels maybe 20 m before it hits the ground, if you shoot a bullet it may travel a mile before it hits the ground, if you are able to get a high enough velocity would make the object travel half way around the earth before hitting the ground and even higher velocity the object would travel all around the earth, which is an orbit.  If there was no gravity this high velocity object would not orbit instead it would go in a straight line and leave earth behind, but since there is gravity it doesn't go in a straight line, it falls toward earth making a curve instead of a straight line.  Since the object is continuously falling it is just like the elevator falling where someone in the elevator was weightless.
So that means that astronauts on the ISS are weightless even though they are in a gravitational field.  To find out how high the gravitational field is we can use Newton's law of universal gravitation to calculate the acceleration due to gravity.

g = acceleration due to gravity
G = gravitational constant
m = mass of the earth
r = distance from the center of gravity

For the surface of the earth the acceleration due to gravity (g) = 9.8 m/s^2
For the ISS, at an altitude of 408 km the acceleration due to gravity (g) = 8.2 m/s^2
So a 70kg astronaut would weigh about 686 N or 155 pounds on earth.
A 70kg astronaut would weigh about 574 N or 129 pounds on a stationary platform at the altitude of the ISS.  The astronaut would weigh less but certainly not weightless.

If the earth was all alone in space how far away from earth would you have to be to become weightless?  Well the equation says you will basically never reach zero, so let's just pick a really small weight, like .05N or about 1/100 of a pound.

So for a 70 kg astronaut that would be an acceleration from gravity of .05/70 or 7.14 x 10^-4 m/s^2.
To find the distance where the acceleration was 7.14 x 10^14 m/s^2

or

So the distance from earth (r) to be weightless (1/100 pound) is about 750,000,000 m or 750,000 km from earth.  Which is about 2 times the distance to the moon.