"feel this force"? What kind of physics is this?
The first and obvious thing to note is that the force exerted by your seatmate is very small compared to other forces acting on your body. Now, the sum of all forces is zero assuming you are not accelerating, so it is not clear that you could feel something even if the force was not very small.
So, let's first simply consider the gravitational force exerted by the Earth on your body. Can you feel that? The perhaps surprising answer is no. Gravity acts uniformly throughout your body and does not cause stresses to build up. This means that you cannot feel the difference between floating in deep space or being in free fall in Earth's gravitatonal field. It feels exactly the same.
When you sitting on your chear, what you can feel is the normal force exerted by the seat on your body. The atoms in the seat push against the atoms of your body that are in contact with the seat. This causes your body to deform slightly and stresses build up in your body. Equilibrium is reached when due to the internal stresses on every part of your body a net force of zero acts (the force due to the stresses plus gravity is zero).
These deformations in your body is what you can feel and these are due to the normal force and not gravity.
So, what about the gravitational force exerted by your seatmate. That's far smaller than the force exerted by the Earth, so it seems that you cannot feel that at all. However, because you are so close to your seatmate the force exerted by your seatmate on you is not constant accross your body. This nonuniformity in the force does lead to stresses in your body (we call this a tidal force).
Now, in case of the Earth's gravity, there will also be a tidal force that directly contributes to the stresses that build up in your body. In tidal force due to the your seatmate can be roughly estimated to be 100 times smaller. Both these effects are swamped by the effects you feel on your body due to the normal force.
becos the force is very small i.e.,negligble.
hence the G has its very small value