Infinities in Newtons law of gravity (for point particles) Newtons law of gravity for two particles of mass $m_1$ and $m_2$  is:

$G\frac{m_1.m_2}{r^2}$. 

Supposing that the particles are point particles then gravitional attraction will bring them closer together, and in fact infinitesimally closer together. Now in Newtons time there was no theory, as far as I am aware of inter-atomic forces that would have kept these two particles apart, so the gravitional attraction is asymptotically infinite. This is nonsensical, and either one can say that point particles cannot arbitrarily approach one another, or that particles can never be point particles and must have extension - this in fact includes the previous solution, as the notional point positions of the centre of mass of a particles with extension cannot obviously approach one another.
In Classical Mechanics, would this have counted as evidence of either particles cannot be point masses; or of some then unknown repulsive force that acts at very small distances.
What does the historical record show?
 A: I think the first few sentences of Landau's Mechanics puts it elegantly:

One of the fundamental concepts of mechanics is that of a particle. By this we mean a body whose dimensions may be neglected in describing its motion. The possibility of so doing depends, of course, on the conditions of the problem concerned.  For example, the planets may be regarded as particles in considering their motion about the Sun, but not in considering their rotation about their axes.

This speaks to the notion Newton and his contemporaries had in mind.  Newton in particular seems to be fairly vague about his notion of bodies that appears throughout the Principia, but Corollary IV gives a hint to his thinking:

The common centre of gravity of two or more bodies does not alter its state of motion or rest by the actions of the bodies among themselves; ...  And therefore the same law takes place in a system consisting of many bodies as in one single body, with regard to their persevering in their state of motion or of rest. For the progressive motion, whether of one single body, or of a whole system of bodies, is always to be estimated from the motion of the centre of gravity. 

But in general, it seems Newton wasn't too explicit in some of the subtleties.  According to Essays in the History of Mechanics - Clifford Truesdell, it was Euler in his Mechanica that pointed out some of the subtlety.

... while Newton has used the word 'body' vaguely and in at least three different meanings, Euler realized that the statements of Newton are generally correct only when applied to masses concentrated at isolated points; he introduced the precise concept of mass-point and this is the first treatise devoted expressly and exclusively to it.

In particular, the relevant parts seem to be in Volume I, Chapter 2 of Mechanica.  Here I try to pull out relevant parts, where he builds up the idea of replacing a body composed of many parts by a single point at its center of mass, and a set of forces of restitution imagined to be infinite elastic forces keeping the various parts of the body joined together.

174 ... it appears possible to determine the motion of a small body, acted on by any kind of forces. ...  175. The force of restitution is that imaginary infinite force, which restores the separate parts of the body again to their previous state. ... 177. ... the restoring force must be considered as provided by an infinite elastic force ... 182. Therefore although the force of restitution is imaginary and only exists in the form of thoughts, yet the effect of this follows the real laws of motion. ... 184. ... that bodies separated into any number of parts can be brought together at the common centre of gravity.

So, not to put words in the mouths of the likes of Newton and Euler, but it would appear as though the talk of point masses going as far back as the beginning is much in line with the quote of Landau's I opened with.  It was considered a useful simplification of problems where the extent of the body was small compared to its motion more generally.  For gravity in particular, Newton (and Euler) spent great efforts demonstrating that for an extended body, one could replace its individual pieces with a point at the centre of mass, without affecting the analysis.  And while at the time there was not a precise theory of any forces that could be keeping those bodies together, preventing their collapse under gravity alone, they had no difficulty imagining those forces as infinite elastic forces between the individual pieces.
A: Classical mechanics has only theoretical point masses. In the simplest case of taking the particle's dimensions to a point, the following argument would hold.
The mass of a particle would be given by its mass density times its volume.
Take the gravitational field:
$$
{\bf g}({\bf r}) = -G\frac{m_1}{|{\bf r}|^2}{\bf \hat{r}},
$$
$m_1$ will be proportional to volume times density  of the material the particle is made of.
The volume is proportional to $r^3$, therefore the infinity at $r=0$ is avoided; there will be 0 mass there. The force  between $m_1$ and $m$ goes like:
$$
{\bf F}({\bf r}) = m {\bf g}({\bf r}),
$$
so it will be zero there as well.
A: I believe that in Physics, when introduced to a law, we must first ask what observation  does it predict. In the case of the gravitational two body problem, we have a Hamiltonian
$H = \frac{p_r^2}{2\mu} + \frac{L^2}{2\mu r^2} - \frac{G \mu M}{r}$,
where $\mu=m_1 m_2/(m_1+m_2), M=m_1+m_2$, $L$ is the total angular momentum, and we have done a transformation into the centre of mass. There is then the so-called effective potential 
$V_{eff}=\frac{L^2}{2\mu r^2} - \frac{G \mu M}{r}$,
which is always repulsive as we approach $r \to 0$ if $L \neq 0$. An intuitive interpretation of this is simply that the Newtonian potential isn't diverging "quickly enough" and the particles always miss (up to a single class of situations "of measure zero"). So most of the time we don't have to bother with the divergent force. From a strictly physical point of view, this was a sufficient description of the solar system where surely $L \neq 0$ and for gravitation on the Earth.
For the rest, I think the historical review of alemi is better than anything I can produce. Just a note on the "infinite elastic forces", in formal calculations these were realized just by rigid constraints such as $|\vec{r}_1-\vec{r}_2|=l$ or for example for "hard spheres" $|\vec{r}_1-\vec{r}_2|<2d$. There was no observational probe of the micro-structure, so "elementary particles" could have as well been hard spheres and this was indeed a frequent insight in the micro-structure.
Last note, in the case of planets I recall a calculation carried out by Newton which actually went to great lengths to prove a variant of Gauss's theorem by proving that we don't feel the pull of the spherical massive shell inside it and thus the gravitational pull does not diverge inside planets. The calculation uses a continuum formulation as stated here already by auxsvr. 
This shows that until maybe the beginning of the twentieth century, when all the experiments of Millikan, Rutherford, Perrin and others made this further impossible, one could easily overcome the point-particle conundrum just by considering continuous matter.
To conclude, point particles are still a problem today in Quantum field theory, where the elementary particles are considered as point particles and infinities also arise at wrong places. In general relativity, the gravitating point particle is classically (i.e. not in quantum gravity) a tiny black hole which makes the problem even worse. The problems with point particles can be e.g. understood as the core of one of the issues leading to the postulation of strings as elementary particles instead of points. For a bit more on this topic, you can read about renormalization on wiki.
A: The infinities of Newton's law of gravity as well as Coulomb's law cannot be removed, as far as I'm aware. However, they needn't be removed for distributions of charge/mass, because the integral $$\vec{F} = -Gm\int_D  \frac{\rho(\vec{x}') (\vec{x} - \vec{x}')}{| \vec{x} - \vec{x'}|^3} d^3x'$$ converges even the integrand diverges at a point. This is easy to see if you convert it into spherical coordinates and $\rho$ is continuous, but this conclusion also applies in the case that $\rho$ is merely bounded, cf. Kellogg, Foundations of potential theory, 1929.
A: There are problems mathematically with point particles, pointy surfaces, and the like. Point particles can be made to go zooming off with infinite velocity in finite time. Systems that violate the Lipschitz conditions can be set up, creating non-deterministic classical mechanics problems.
These are conceptual problems. We use point particles because real objects with a spherical mass or charge distribution look just like a point particle from the outside. (From the inside, they don't.) When those real particles get too close together they collide. Singularity averted. The same goes with those systems that violate the Lipschitz conditions. When you look at what appears to be a pointy surface, it's not. If you have to look extremely close, it's time to turn things over to quantum mechanics.

Papers that describe the mathematical issues described above:
Norton, J. D. (2008). The dome: An unexpectedly simple failure of determinism. Philosophy of Science, 75(5), 786-798.
Saari, D. G., & Xia, Z. J. (1995). Off to infinity in finite time. Notices of the AMS, 42(5).
Xia, Z. (1992). The existence of noncollision singularities in Newtonian systems. Annals of Mathematics, 411-468.
