Difference between Free Fall and Constant Velocity We know that astronauts in the ISS feel "weightless" because they are in a perpetual free fall. The earth's gravity is the only force acting upon them, and it is accelerating them towards the earth's center at the same rate as the ISS.
However, now imagine a crew in a spaceship very far from any planet, star or massive object. So much so that there isn't any gravitational force acting upon them. The spaceship is also traveling at a constant velocity, so the crew has a zero acceleration.
In the two situations, the astronauts feel "weightless" and are "floating" in the same way. However the two situations are very much different in terms of forces and acceleration. How can we explain that?
 A: You can say the net force on the astronauts is zero for both cases. While they are orbiting the Earth, the centrifugal force matches the gravitational force to make the net force zero, and in empty space there are no forces.
The centrifugal is usually said to be a 'virtual' force, but it is very real in cases like these. When we consider such a system in classical mechanics we usually don't consider observers being inside the objects that's in orbital motion. To such an observer, the centrifugal force is real. To an observer looking from the outside, the centripetal force is real, one of them pulls the object in, the other pushes the observers inside it out. Since the 'weightlessness' happens to the observers inside the orbiting object, the centrifugal force is a valid way of explaining this.
A: The feeling of weight is just the feeling of "something" pushing on you. For example, stand in an elevator accelerating upwards, and you will feel heavier. Stand in an elevator accelerating downwards, and you will feel lighter.
In both scenarios you describe it is the case that there is nothing pushing on you to cause your acceleration. On the ISS you are at rest relative to the ISS, so nothing is pushing on you. If you were moving at a constant velocity with no forces acting on you, the same thing applies.
Note that this is related to the equivalence principle, which you might be interested in reading about.
A: 
In the two situations, the astronauts feel "weightless" and are "floating" in the same way. However the two situations are very much different in terms of forces and acceleration. How can we explain that?

Welcome to SE, and excellent first question!
Your question is very much along the lines of what led Einstein to the equivalence principle, except that there were not any astronauts or space stations in his day, so his feat of imagination was all the more impressive. What he and you realized is that, on a small scale like inside the ISS, the force of gravity is indistinguishable from an inertial force (also called a fictitious force or a pseudo force) that arises from using a non-inertial reference frame.
That implies that an inertial frame is one that is in free fall. There is no experiment that the astronauts can perform entirely onboard (meaning without getting information from outside) that will distinguish the two situations. Taking this idea seriously leads to general relativity.
Basically, gravitational forces and inertial forces share the following properties:

*

*they are proportional to mass

*they can be removed through choice of reference frame

*they cannot be detected with an accelerometer

In particular, the third property explains the astronauts inability to distinguish the two situations. Physical sensations that are normally associated with gravity are, upon closer analysis, actually due to some other force. Like the contact force from the floor or a chair.
A: 
In the two situations, the astronauts feel "weightless" and are "floating" in the same way. However the two situations are very much different in terms of forces and acceleration. How can we explain that?

I disagree; I think the two situations are almost identical in terms of forces.
In the case of the ISS, there are practically no forces exerted on the space station other than gravity, and this produces the sensation of weightlessness. In the case of a spaceship in deep space far away from everything, once again, there are practically no forces exerted on the spaceship other than gravity, and this produces the sensation of weightlessness. So, in both cases, the crew feels weightless for exactly the same reason.
The only difference between the two situations is that the ISS is strongly pulled by gravity, whereas the spaceship in deep space isn't pulled by gravity at all. However, this doesn't have any effect on what the astronauts experience. Ironically, the force of gravity doesn't make us feel like we have weight; that feeling is caused by all the other forces besides gravity.
