I recently learned about gravitational force and found out the equation for gravitational force on a object by an object according to Newton's law of universal gravitation.
$$ F = \frac{Gm_1m_2}{r^2}. $$
$r^2$ denotes the square of separation between two objects. So the question is what happens when this separation is very small. For example when someone is standing on ground, separation between him and Earth is very low (zero).
Does gravitational force $F$ becomes ∞ then?
The $r$ value in this equation represents the separation of the two bodies' centers of mass.
So, when you're standing on the surface of the earth, then the value of $r$ is equal to $r_E$, the radius of the earth, which is $6378$km or $6.378\times10^{6}$m.
To answer your more general question, the force between two masses does, indeed, increase as separation decreases, but as separation gets smaller and smaller, other forces start to dominate.
The electrostatic forces between molecules and atoms are millions of times stronger than the gravitational forces between them.
The strong nuclear forces between nucleons are even stronger still.
Still, if you have enough mass, gravity can overcome all of these forces. When a large enough star collapses to an ever-smaller point, you get a black hole. While Physics tends to try to avoid talking about infinities, the nature of a black hole is such that you can, in theory, get arbitrarily close to its center of mass.
However, at this point, you are lost to the universe and what happens inside a black hole, stays inside a black hole. Physicists are not really sure what happens there.
There are various issues with this thinking, I will ignore the issue of the radius not being 0 on the surface, as you covered this already.
1: While space is continuous over R, physics as a whole tends to break apart and collapse at a non-infinitesimal size, that being the Planck Length, which is around 1.6 x 10-35 metres. Beyond that we really don't know how things work.
2: At small enough distances, even before you reach the Planck Length, other forces take over. So gravity being considered as the primary force. Gravity becomes negligible. It is actually very weak in general, to the extent that there are many theories which attempt to understand why it is in such a completely different league than the other fundamental forces.
3: As also mentioned before, it is based on the distance to the centre of mass, which grows very, very small as you get to very close distances, as the object would have to become less massive in order to be smaller without collapsing into a black hole. This tends towards 0 as you get closer.
Sorry if I poorly explained anything.
Although the question is interesting, the example you took seems to be a spin off. I will address the physical example first to show why gravitational force doesn't really blow up when really close:
As you know distance is measured between the centers of the masses. Can this distance be very close to zero?
Actually no because mass would become zero as well! Think of it. Mass exists in matter and matter has volume, hence to have significant mass you at least require some volume (even if you take the densest material in the universe). Now lets take your example of a man standing on earth. Since the distance is measured from the centers, here the distance is actually slightly greater than the radius of earth (6300km)! Okay, so now you decide to reduce this distance and dig a tunnel towards the center of earth, what happens then? The mass of earth that contributes to the gravitational force changes and is lesser than the actual mass!
The effective mass of earth is given by (assume earth to be spherical and the density to be same everywhere on earth): $$m_{eff}^{earth}=M_{earth}.\frac{r^3}{R^3_{earth}}$$ where $r$ is distance between the man and the center of earth. Clearly as $r\to0$, $m_{eff}^{earth}\to0$ faster than the denominator ($r^2$). Hence physically is not possible to make the gravitational force blow up by decreasing the distance.
Now, why is your question interesting?
It is interesting because this is exactly the dilemma that physicists face when they want to study the small scale/distance effects of gravity (they just cant go to small scales without making the gravitational force negligible), in fact the dilemma is so huge that we still dont have a microscopic theory for gravitation. The main reason for this being what you pointed out together with the fact that at small scales other fundamental forces hugely dominates any gravitational effects, making the gravitational force negligible.
