Astronauts on the International Space Station feel weightless because

Question

Their apparent weight is zero.
They are above the earth’s atmosphere.
The gravity of the earth and the sun are directed opposite each other, so there is no net gravitational force.
They are far enough above the earth’s surface that the gravitational force is negligible.

Answer 1

Their apparent weight is zero.

 

They are in a constant free-fall (that is, there is no resistance to the force of gravity's pull).

 

Imagine a rocket that goes straight up then falls straight down.  There are G-forces (that is, times the pull of gravity) when going up and small resistance (only air friction) when falling back to earth.

 

Now, shoot the rocket on an angle.  It still has G-forces going up and almost no resistance falling down.

 

Finally, shoot the rocket on such an angle that, when it falls down, it has gone past the earth;  and keeps going past the earth; and keeps going past the earth ...  It is in orbit !! [note: The earth rotates west-to-east, so why not shoot the rocket in that direction to take advantage of the sling-shot effect?]

 

The ISS orbits at 205-to-270 miles above the earth at a speed of 17,200 miles per hour [which means 15.54 earth orbits per day].  This is still "low earth orbit."

 

At that altitude above the earth, the sun has sufficient gravity to keep the massive earth in orbit, but doesn't pull the ISS very much.  The earth's gravity is still strong enough for a large mass to fall quickly (and, therefore, to burn up in earth's atmosphere).  However, the orbital speed of the ISS keeps it falling at a roughly constant distance from the earth (thus, constantly missing the earth at it falls -- this is called "an orbit.")

Answer 2

Their apparent weight is zero.