Do you feel gravity? I have been reading a few articles about the question why we don't feel/notice gravity in everyday life, but I couldn't understand why exactly we don't feel/notice it, that is, why we don't feel a strong force pulling on us at every moment.
 A: Theoretically, the only unambiguous way to "feel" gravity would be to travel near an extremely strong source of gravity, or rather an extremely strong gradient of gravity (e.g. a black hole), and feel tidal forces from such a massive object.  (that, or gravitational waves from a binary system of black holes)
The rest is a matter of definition.  We could define "feeling" gravity as perceiving its effects on our bodies.  And the way we perceive the consequences of gravity on Earth is a sensation of the ground "pushing" upwards on our feet, and preventing us from freefall towards the center of the planet.
However, don't forget that gravity is basically curvature in spacetime.  When you're in freefall, you simply follow a geodesic through the curved spacetime in your vicinity.  In this sense, you can't "feel" gravity because there's nothing to be "felt" -- in the absence of other forces, your body is simply following a geodesic that is curved by the presence of a massive body (the earth).
If you were inside a free-falling box, you would never be able to tell whether you're in deep space (far away from any massive objects), or plummeting towards the surface of the earth.  So, in this sense, gravity cannot be "felt."
A: Probably this opinion means that when feeling gravity one actually feels reaction force. For example, while standing you feel pressure of the ground upon your feet. 
But when you fall down you feel weightlessness, while you are under action of gravity in fact. 
This is because gravity acts on all your body's cell or atom equally.
These are all geek's points.
A: This will be concise:


*

*Gravity is the weakest of the four fundamental forces. You will feel electrostatic shock and you will feel the magnets attracting/repelling, but you won't feel the attraction between you and your neighbor ($F_{gravity}<<F_{electromagnetic}$).  

*The fact that you can say some things are heavy and other things are light is the proof that you feel gravity, because weight is proportional to gravity ($F_{weight}=mg$).  

*If you are referring to the fact that there is a gravitational attraction between us and the earth, you do indeed feel it. You feel through the normal force, the force of the ground or bed or desk pushing back up on you... if there was no gravity you would float away and then you wouldn't be in contact with anything. More to the point, stick your arm out... you won't be able to hold it there for long, your muscles will get tired and you'll put it down. Those muscles were counteracting the gravitational-pull downward.

A: You don't 'feel' gravity because your nervous system is differential.
Your body measures contrasts and adapts to constant values. Since you're always experiencing the same gravitational force, your sense of 'down' is constant and not reported to your brain as anything to take notice of. Take off in a plane or ride a roller coaster and suddenly your nerves report the change.
This is a neurophysiology question as much as a physics one.
A: The geometry of spacetime dictates that objects under the influences of no forces will follow certain trajectories--"free-fall" trajectories, called geodesics.  Gravity alters the geometry of spacetime, changing the trajectories of geodesics, but gravity alone cannot make an object follow a non-geodesic path.  The uniqueness of geodesics is why we consider motion only under gravity to be the absence of forces.  That is, in relativity, we do not consider gravity to be a force.
On the other hand, electromagnetic forces and such are still forces, detectable and measurable in a way that multiple observers can agree upon.  Hence, they can be "felt."
All in all, this concept is merely an extension of an idea from Newtonian mechanics:  that objects under forces follow straight paths.  When spacetime can become curved, the notion must be modified, as straight paths are no longer natural (consider, for instance, lines of longitude on a sphere, which seem straight and parallel to one another at the equator but intersect at the poles--does that make them no longer straight?).  Free-fall paths are the analogous notion in relativity.
It is my contention that this is what's commonly meant--that gravity does not contribute to four-force.  There are other aspects to this problem as well, several of which that don't require relativity.  The near-uniformity of Earth's gravitational field near its surface means that, were you in free-fall with a collection of objects, how would you even really know you were falling?  They would all accelerate the same way you do, after all.
Conversely, a rocket accelerating in empty space versus sitting on the ground on Earth can't be distinguished.  This is exactly one of the thought experiments used to commonly discuss the problem, and note that a rocket accelerating accelerates forward, while a rocket sitting on the ground appears not to accelerate, but to maintain the symmetry of the situations, we must conclude it is actually accelerating upwards.  Upward is the "real" force we're being subjected to.
Yes, the human body is attuned to being under the influence of an upward force, but the question is more one of telling the difference between being stationary in outer space, in the absence of gravity, vs. moving freely under gravity's influence with nothing to stop you.  These two cases cannot be distinguished, really, without making reference to some frame of reference that you decide must be fixed (a planet, background stars, etc.).
A: Yes, you can get an intense feeling of gravity when you spend enough time in a zero gravity environment.
For example. You could get inside a pool for an hour or more and then get out of it to experience the pull.
A: You feel the absence of it when in elevator or on a swing. Maybe they meant you are so used to it you don't consciously feel it. You are built to this particular gravity. Same way a car don't think about its brand or a more famous example a fish doesn't think of water. 
A: Forces move objects in spacetime, and due to inertia you feel this. Gravity does not move objects in spacetime (hence it's not a force), instead gravity warps spacetime and you will warp with it provided nothing gets in your way, when something gets in your way (the ground at your feet), then you feel it.  The ground at your feet is accelerating upward due to the force of compression caused by the warping of space--that is the force you feel.
