What's the reason it behaves differently from all other forces? What I mean is, if you're in orbit you're accelerating toward the earth at almost 9.8m/s^2, but you feel nothing. If you are riding a space ship going for a slingshot orbit around the Sun, you are accelerating at more than 28Gs, but still feel weightless. But if you're just standing on earth, the ground pushes pack against gravity and you feel it. Opposingly, if you are in a racecar accelerating at 1G, you feel it. Because you have mass you resist the acceleration and you feel it. But mass has no resistance to acceleration due gravity. Why?
From a Newtonian perspective, the difference between being accelerated by gravity in freefall (which includes orbits) and being accelerated in a car has to do with the fact that you only "feel" accelerations when the external force is only being applied to one part of your body, rather than accelerating every particle equally as with gravity. For example, if you're sitting in a car the only external force is on the part of your body that's in contact with the seat, so if the rest of your body accelerates too, it must be because of internal compression forces within your body.
It's easier to understand if we replace a complicated human body with a spring that has the back end attached to the seat. Then the force from accelerations will initially only cause the back end of the spring to accelerate while the front end does not, causing the distance between the back end and front end to decrease, and by Hooke's Law this creates a force that tries to push the two ends apart, so this begins to accelerate the front end in the same direction as the back end. If the acceleration is constant, then after some oscillations an equilibrium will be reached where the distance between the front and back end is shrunk relative to the relaxed length, and it's shrunk by exactly the right amount so that the spring force on the front end accelerates it at the same rate as the back end is being accelerated by the net force on the back end (which is a sum of the force from the seat in the forward direction, and the backward spring force).
So basically, the same thing happens with your body--when the seat accelerates your back side, your body tissue compresses a bit until the spring-like force is accelerating your front side at the same rate as your front side, and this compression is what you feel physically as acceleration. In contrast, with gravity all parts of your body are being accelerated equally so there is no compression, and you don't feel anything. Note that the reason gravity is different here just has to do with the fact that we don't normally encounter other external forces which act uniformly on all parts of our bodies, but this would certainly be theoretically possible--for example, if your body had a net negative electromagnetic charge, and the extra negative charge was distributed uniformly throughout your body, then if you were being accelerated by electromagnetic attraction to a body with a net positive charge, you wouldn't feel this either (at least not unless you were close enough to the positively-charged body so that the electromagnetic attraction of the nearer part of your body was noticeably stronger than the attraction of the farther part, which would be the equivalent of a tidal force in gravity).
If you analyze things using general relativity (Einstein's theory of gravity) rather than Newtonian gravity, in this case gravity is not treated as a "force" at all, but rather an effect of the fact that matter and energy create curvature in spacetime, and objects in freefall (those with no net non-gravitational forces acting on them) naturally follow geodesics in the curved spacetime. In Newtonian physics, you can (and usually do) analyze problems in a global coordinate system known as an inertial frame, and objects in freefall in a gravitational field are treated as moving non-inertially (not a constant velocity relative to inertial frames). In contrast, in general relativity you can only have "local inertial frames" in infinitesimally small regions of curved spacetime, and in such frames objects in freefall are the only ones moving inertially, whereas something like an object sitting on the surface of a planet is moving non-inertially (being accelerated by the normal force from the ground, as measured in the local inertial frames of freefalling observers). See this article on the equivalence principle for more on these ideas.
The answer from a Newtonian perspective:
TL;DR: Objects do feel gravitation, but only if they're very big, or if the gravitational field is very strong.
Suppose you are in a spacesuit and are orbiting the Earth. Your feet are pointed toward the Earth, your head into space. Because gravitation is a 1/r2 force, the force on your feet is slightly stronger than the force on your head. Gravitation is trying to tear apart! This is a very small force for someone a couple of meters tall. You can't feel it. Now imagine straightening out your arms so they are perpendicular to your body. The force on your fingertips is downward but also a bit toward your body. Gravitation is trying to squeeze you from side-to-side. Here's a diagram of these forces on a spherical object:
The tidal forces exerted by the Moon and Sun across the Earth are rather small. They are immeasurably small for an object the size of a person.
What we do feel are forces that cause the sensors in our ears to move relative to our bodies (gravitation doesn't do this) or cause tension or compression in our bodies (once again, gravitation doesn't do this). Every force other than gravitation does do this. We feel everything but gravitation.
The answer from a relativistic perspective:
TL;DR: Gravitation isn't a real force in general relativity.
We don't feel the fictitious centrifugal force or the fictitious coriolis effect. Unlike the very real force you feel when your buddy offers to give you a ride in his new Corvette, those fictitious forces aren't "real" (hence the name). The answer to this question from a relativistic perspective is very simple: Gravitation is a fictitious force in general relativity. You can't feel it because it's not "real". For more, read up on the equivalence principle. There are a number of questions at this site that are tagged with the equivalence-principle tag. Look at those questions and the answers to them.