Why are there so many fundamental differences between contact forces and gravity? I had this very fruitful conversation about the inertial motion of charged particles on gravitational/electric fields.
A field force like gravity, can't be felt, it does not produce proper acceleration, nor it does do work on a mass - because its just spacetime curvature you are falling into, and it does not deliver (in first order) energy to you. If you are blindfolded (no external references), you can't tell if you are on a geodesic of the gravity field or standing in empty space, you are just weightlessness, moving in an inertial referential. Gravity produces no internal compression or dilation on you, so your accelerometers measure zero.
On the other hand, contact forces can be measured by accelerometers, they do work on the object, and promote the object to non-geodesical trajectories. Accelerometers are only able to measure contact forces, not gravity force, because gravity is a field force that acts on all of you at the same time. Contact forces on the other hand act layer by layer from the contact surface on through the body being properly accelerated.
But contact forces are in ultimate analysis are just electromagnetics forces between the atoms of two distinct bodies of matter. I understand (see here) that everything that carries energy and/or momentum do produce a curvature in spacetime.
So, both masses and charges$^\star$ produce spacetime curvature, but we see gravity as a force field that acts on whole bodies at once (and thus cannot be measured by accelerometers), while electrostatic force acts like a contact force we see acting on layer by layer of a material, acting to promote proper acceleration, and thus renders objects to non-inertial frames.
So, what is the fundamental differences between these two? Is it the fact that gravity has a much larger range than the electromagnetic force. Or is it some more fundamental reason?

$^\star$ Charges produce spacetime curvature indirectly due to both their masses and the electromagnetic fields they generate.
 A: The difference is that all masses respond to gravitational fields in an identical way, while the response to electric fields depends on charge.
This means that for a nearly-constant gravitational field (like is found near the surface of the earth), each portion of the object will have nearly identical accelerations.  With all accelerations the same, the gravitational field doesn't create distortions in the object, nor are any additional internal forces necessary to maintain the objects shape.  This is another way of saying that a nearly constant gravitational field can't be "felt".
For an electric field the situation is different.  Most objects are approximately electrically neutral.  So far from a charge or an electric field, there is no force induced.  But as objects get very close, the electrons in the atoms get close enough that the forces between them are no longer shielded.  The forces induced are felt differently at different portions of the object.

*

*Gravity is weak enough that we simply cannot feel the forces from small objects.

*We are so far from large objects (like the earth) that the gravitational field is approximately constant across our entire body.

*The positive and negative charges in our bodies have very little reaction to EM fields and charges unless they are extremely close.

*It would be possible to "feel" gravity if you could get a field that changed significantly over the size of your body.  We cannot create them.

A: There are four$^\star$ fundamental interactions that explain all the forces we experience: the gravitational force, electromagnetic force, and the weak and strong nuclear forces. A "contact force" is not fundamental type of force, but an effective description that applies on large scales. Really, you would expect all forces to be measurable by an accelerometer, since $F=ma$. This very much includes the electromagnetic force, which is absolutely a "field force."
The real distinction is not between field forces and contact forces, but between gravitational and non-gravitational forces. What makes the gravitational force special is the equivalence principle. If you are freely falling in a gravitational field, and you do experiments that are "small" compared to the characteristic length scale over which the field changes, then there is no way to measure that a gravitational field is present. From the point of view of general relativity, we would say you are following a geodesic in spacetime. Any other force will cause you not to follow a geodesic, and this is what is measured by an accelerometer.
Finally, I want to caution you about a crucial assumption in the previous sentence: "you do experiments that are "small" compared to the characteristic length scale over which the field changes." This is not always the case in gravity. For example, if you fall into a black hole, as you approach the singularity, the gravitational field at your feet will become much larger than the field at your head, and you will be spaghettified. So it is not generally true that gravity only acts on "you" as a single object. In fact this gets the logic exactly backwards; the real statement is that you will follow a geodesic when you happen to be small compared to the length scale over which the gravitational field changes.

$^\star$ Arguably we should add a Higgs force to this list as well.
