Why does a free-falling body experience no force despite accelerating? Note: For the purposes of my question, when I refer to free fall assume it takes place in a vacuum. 
From my (admittedly weak) understanding of the equivalence principle, falling in a gravitational field is physically indistinguishable from floating in interstellar space. This would make sense to me if gravity simply caused an object to move at a constant velocity. Moving at a constant speed, or floating in space, are just two different ways of describing an inertial frame, and are fundamentally no different. But free falling in a gravitational field means accelerating continuously, and doesn't an accelerating body experience a force? Then isn't free falling fundamentally different from floating in space?
 A: Before telling you why an observer in free fall does not feel any force acting on him, there are a couple of results that should be introduced to you.
Newton's second law is only valid in inertial frames of reference:
To measure quantities like the position, velocity, and acceleration of an object, you need a coordinate system $(x,y,z,t)$. Now the coordinates that an observer in uniform motion(constant velocity) uses is what we call an inertial frame of reference, and the coordinates of an observer in non-uniform motion is what we call a non-inertial/accelerated frame of reference
Now $\vec{F}=m\vec{a}$ is only valid in inertial frames of reference. This means that only observers in uniform motion are allowed to make valid inferences about an object being acted upon by a force(and hence being accelerated) and so on, while non-inertial frames of reference are not justified in making inferences about an object being accelerated or not.
Fictitious forces
For example, consider the case of two observers, one who is at rest on the ground and the other who is in an accelerated car(say moving in the positive x-axis with constant acceleration) that is passing by the observer resting on the ground. The observer in the car will discover a very peculiar situation in his frame of reference, when he holds his medal by a string, he immediately observes that the medal starts to move backwards in the negative x-direction and the string that is holding the medal makes an angle with the vertical. If he has a ball in his hand and lets it go, he observes that the ball starts to accelerate backwards(negative x-direction) until it hits the back of his car. So it appears as if there is some mystical force in this observer's frame that has no obvious origin, which acts upon all objects and accelerates them backwards. This observer will further note that this mystical force is proportional to mass, or in other words, the acceleration of any object is independent of it's mass, so that if you hold two different masses in your hand and let them go, they will hit the back of the car at the same time.
But the observer who's at rest on the ground will object! he will argue(rightly) that there is no mysterious force that is accelerating the objects in the car. The fact that any object "appears to accelerate" backwards is a simple consequence of these two following facts:
1)The car is accelerating onward in the positive x-direction.
2)The objects, when they are let go, they are moving with constant velocity(they both have the same velocity) in the positive x-direction according to the ground observer, and following Newton's first law, they will continue to do so, but the car is still accelerating onward, so they eventually hit the back of the car at the same time.
So as you can see in the above example, an observer in accelerated frame, when making inferences about an object being accelerated or not, will arrive at the wrong conclusions, since Newton's law are only valid in an inertial frame. If he makes inferences, he concludes the existence of some fictitious force with no obvious origin, that is proportional to the mass, but this is just an artifact of the observer being in an non-inertial frame and using Newton's laws to make inferences about the motion of objects. This fictitious force can simply be explained due to the combined result of the acceleration of the car and the inertia of the bodies inside the car that were just let go.
(There is one complication which I ignored in this example, namely gravity, actually when the masses are let go inside the car, their trajectories are not going to be straight lines, but sections of a parabola, But if you performed the above example in space-free gravity, the example holds exactly true). 
The litmus test for an inertial frame of reference 
Newton's first law is the litmus test to tell apart whether you're accelerating or not. If you are floating in space, and there is an object in your hand, and you let it go(at rest), it will stay at rest. But if you're accelerating(like the case of the car), and let the mass go, it will start to mysteriously accelerate with a force that is proportional to the mass.
Einstein's big idea
The fact that gravity has no obvious origin, and is proportional to mass, prompted him to suggest that maybe gravity is just another fictitious force, that results from us, observers who are at rest on the ground being in an accelerated frame of reference.
But to ultimately prove this is true, he had to find a frame of reference in which this force of gravity disappears, just as we concluded that the mystical force in the car's frame of reference is fictitious, by switching to the frame of reference of an observer who's standing on the ground.
And Einstein found such a frame! Switch to a freely falling frame of reference and this mystical force of gravity suddenly disappears; you feel weightless. Put a scale down at your feet and it will read zero. Try to hold a ball by a string that is attached to your hand and the tension on the string immediately disappears, and it becomes loose as you start to freely fall, and so on. In such a frame there's no force of gravity, just as there is no mystical force when you switch from the car to the ground frame of reference.
Newton's explanation
Newton will argue that gravity is not fictitious, but real. The fact that you feel no force acting on you when you're in free fall can be explained like this:
According to Newton, an observer in free fall is being acted upon by the force of gravity, so he's accelerating, so his frame of reference is not inertial and any inferences he makes about motion using Newton's laws are incorrect. Since the freely falling observer is accelerating, in his frame, there appears a fictitious force that acts on him upward and is proportional to his mass, but gravity acts on him downward and it's proportional to his mass as well! Therefore, they will cancel each other out, and he feels no force, even though he's accelerating!
Einstein responds
Einstein used the litmus test to tell whether, being in free fall, one is in an inertial frame of reference or not. You hold a certain mass in your hand, and let it go, and it stays at rest with respect to you. This case is totally equivalent to the observer floating in space we described above.
One the other hand, if you're on the ground holding an object and then let it go, it does not stay at rest, but rather it starts to accelerate downward with a force that is proportional to its mass. This case is totally equivalent to the case of an observer in a car letting go of masses that we described above.
He called this the equivalence principle.
So yeah, gravity is indeed fictitious.
Now after you have been introduced to the relevant concepts, the response to your assertion that "But free falling in a gravitational field means accelerating continuously. And doesn't an accelerating body experience a force?" is something like this:
To draw valid inferences about acceleration of any object, you have to be in an inertial frame of reference, otherwise you're led to the wrong conclusions, just as we demonstrated above. Your assertion that a body in a gravitational field is accelerating, and hence should experience a force, is false in the Einsteinian sense. That is because, as we have noted above, you made this assertion being on the ground, and an observer on the ground according to Einstein is in an accelerated/non-inertial frame of reference, so his inferences about a body in a gravitational field being accelerated will be false. Only freely falling observers are justified in making claims about acceleration of objects because they are in an inertial frame of reference.
But even ignoring Einstein and sticking to Newton's worldview, an observer in a free fall experiences no force at all, because gravity(which is real according to Newton) and the fictitious force exactly cancel each other out, even though he is accelerating!
So as you can see, in either cases, whether Newtonian or Einsteinian, an observer in free fall does not feel any force acting on him.
Credits should be given to this video. 
A: Well, everything depends on what you mean by "to experience a force". I suspect that you are thinking of some psycho-physical idea. Indeed both floating in space and freely falling we perceive similar sensations. The reason is simply due to the fact that, in both situations, all particles of our body moves with the same speed (due to a spatially uniform acceleration and the fact that inertial mass and gravitational mass coincide, if freely falling) so that, roughly speaking, the distances between various parts of our body remain constant and we do not perceive internal stresses. We would perceive some force sensation if the gravitational force were not uniform (tidal forces), because different parts of our body would move with different velocities, distances would varies and stesses would take place to hold together our body. 
A: 
falling in a gravitational field is physically indistinguishable from floating in interstellar space

Yes. Indeed, this is one of the founding principles of general relativity and is (one of the forms of) the equivalence principle.
Your argument is that we can feel acceleration, and gravity makes you accelerate, so shouldn't you feel acceleration while you're falling in a gravitational field? Einstein started from the opposite end. He reasoned that you feel no force when you are falling freely, so any theory of gravity must include this basic principle. From that starting point he formulated general relativity.
So if you ask why we don't feel a force while falling freely, there isn't really an answer to that. The universe is just built that way.
I should add that falling freely in a gravitational field is only like floating in space locally i.e. in your immediate vicinity. This is because gravitational fields generally produce tidal forces. You've probably heard that if you fall into a black hole you get spaghettified, i.e. stretched out into a long thin strip, and this is due to tidal forces. So in this case even when falling freely you do feel forces acting on you.
Response to comment:
Newton's first law tells us that if no force (and therefore no acceleration) is applied to an object it moves in a straight line. This is also true in general relativity, but in GR we replace a straight line by a geodesic - where a geodesic is just the trajectory followed by a freely falling object. In GR as long as you are travelling on a geodesic you feel no acceleration.
If you're sitting in an accelerating car then you aren't following a geodesic because you aren't following the trajectory you would follow if the car wasn't there. Actually, even if the car isn't accelerating you still aren't following a geodesic because if the car wasn't there you'd start falling towards the centre of the Earth. The gravitational force we all feel all the time (assuming you aren't falling off a cliff as you read this) is felt because when standing on the surface of the Earth you aren't following a geodesic.
The thing that makes GR hard (well, one of the things) is that geodesics don't necessarily look straight. For example the International Space Station is orbiting in a circle, but actually it is following a geodesic. That's why the astronauts on the ISS feel no acceleration i.e. they are weightless. General relativity basically consists of working out what shape the geodesics are.
A: We don't need to appeal to relativity to explain why you don't feel any force in free fall. Plain old Newtonian mechanics predicts that too.
What you actually feel when you feel a force being applied to you is that the external force applies only to a small part of your body (the soles of your feet if you're standing up and feel the normal force from the floor, or the skin of your back if someone pushes you). If all the force applied to those few of your cells that are in contact with the source, Newton's first law would make those cells would begin accelerating with respect to the rest of your body and you would go out of shape. So the external force sets up stresses within your body, and it's the stresses you can feel.
In more technical language, the thing you can feel directly is momentum flowing through your body, from one place to another.
However, in free fall, there's an external force acting on you -- gravity -- and you're accelerating. However, the nice thing about gravity is that it delivers momentum inside you exactly where it is going to be consumed by making you move, so there will be no flow of momentum within you, no stresses, and nothing to feel.
(When you're standing on the ground, gravity is still pouring downwards momentum into every part of you, but since you're not accelerating, that momentum will have do go somewhere. You can feel it draining into the ground through your feet, and they will eventually get tired).
A: It is incorrect to link the feeling of being accelerated to being accelerated itself. You can be under constant velocity or be continuously accelerated, yet you need not feel anything at all. Let me explain.
The reason you feel compressed or stretched when you are accelerated in a lift is because of the presence of the normal force from the ground on you. The normal force pushes up on your feet while gravity pushes down from your center of mass. That's why your legs feel compressed in an accelerating lift. Your leg is under stress, and that's the feeling of being accelerated.
A free falling object doesn't experience force even though gravity acts on it because there is no other opposing force to induce any stress in your body. In the absence of such a normal opposing force during free-fall, you do not feel anything.
A: There is another aspect somehow overlooked by the other answers. Consider a pile of iron filings accelerated towards a magnet. If you were to arrange so that they all have the same magnetic force per unit mass they would appear to experience no force relative to each other while being accelerated towards the magnet, and if you had weak bonds holding them together they would not brake until the tidal forces got too strong (too close and uniform field assumption is no good).
So the same with gravity.
A: 
Why does a free-falling body experience no force despite accelerating?

Because there is no force acting upon it. If you look at some pictures of the principle of equivalence, you will find that they typically depict a guy in a rocket accelerating through space. There's a force on his feet, he can feel it. They also depict a guy standing on the surface of the Earth. There's a force on his feet, he can feel it. When he's in free-fall, there isn't and he can't.   

From my (admittedly weak) understanding of the equivalence principle, falling in a gravitational field is physically indistinguishable from floating in interstellar space.

Pretty much. Like John Rennie suggested, if you had some sophisticated kit you could in theory tell the difference. 

But free falling in a gravitational field means accelerating continuously. And doesn't an accelerating body experience a force? 

No it doesn't. If you were falling inside a lift, it would feel just like being inside a box floating in space. 

Then isn't free falling fundamentally different from floating in space?

It is. The two situations are not the same. They feel the same, but they aren't. To appreciate this it's better to compare accelerating through space with standing on the Earth. See this where Einstein described a gravitational field as space that was "neither homogeneous nor isotropic". Well, staying still in inhomogeneous space is like not staying still in homogeneous space. But the two situations are not the same.    
A: You need a coordinate system to decide a body’s position, velocity, acceleration, momentum or force on it.  Assume the body is in free fall near the Earth.
1)  First consider a coordinate frame (3 perpendicular rods and a clock) with its origin in free fall near the free falling body. By the equivalence principle we know the rods are falling in unison with the body.  In this frame the position of the body never changes, the body has zero velocity because its position never changes, and zero acceleration because its velocity never changes.  Also, in this frame the momentum of the body is zero and never changes because the body’s velocity is zero and never changes.  Thus in this frame the force on the body is zero. This is the coordinate frame you are using when you are the body and say that you “feel” no forces.
2)  Next consider a frame with its origin in free fall far outside the Earth’s gravitational field and away from the body. In this frame the position and velocity of the body are observed to be changing with time.  The body appears to be accelerating because its velocity is changing with time.  The momentum of the body is changing with time because its velocity is changing with time. Thus in this frame the force on the body is non-zero.
So, when the body is seen to be not accelerating, no force is seen (case 1).  When the body is seen to be accelerating, there is a force (case 2).  It’s a matter of the observer’s coordinate frame.
A: 
CASE -I:Consider the acceleration of a body with mass M when a force of 100 N is applied on a body.
CASE-II : Consider the acceleration of the same body when a force of 1000 N and 900 N respectively are applied on it simultaneously in opposite directions.
The acceleration in both cases will be the same as the net force is 100 N.
Now consider the internal changes in the body. The body in case II is experiencing a much severe crushing power than in case I.
Similarly in a free fall only a single force is acting on the body. 
A: 
Why does a free-falling body experience no force despite accelerating?

False. It experiences gravitational force, which causes an acceleration directed towards the center of mass
However, what you probably had in mind is a body weight. Weight is a force which body acts upon support. Falling body has no support to act upon and thus - no weight. This actually can be best understood trying to measure falling body's weight with a weighing scale. They will show 0 kg, because they are falling itself too and as such have no support too. (Btw, note that a weighing scale actually measures weight- not a mass, that's why you'll get 0 kg for a falling body !)
As a last argument trying to convince you that body experiences a gravitational force when falling - try to reach a ground after jumping from a plane without a parachute. Things which you will experience in last moments will be an effect of gravity force (weight) + momentum.
