Why doesn't matter clump together such that it can't be taken apart again? Given the inverse square law force of gravity shouldn't two particles that are infinitely close to each other be infinitely attracted to one another? For example, suppose the hands of some super deity grabbed hold of two neutrinos and put them infinitely close to one another or suppose that some physicists in a high energy particle physics laboratory shot two neutrinos together at super high speeds. Why should any forces be able to pull the neutrinos apart again after they collide? Why don't we find any tiny super dense lumps of matter around us that are the results of high energy collisions and that we are unable to pull apart?
 A: Ignoring the quantum effects that make such a situation both improbable and overtaken by stronger but shorter acting forces - if we take Newton's gravitational equation with the inverse square law:
$$ F = \frac{G \cdot M_1 \cdot M_2}{R^2} $$
and if, in theory, you get 2 objects to occupy the same exact space - (ignoring quantum and other difficulties there of), The equation only works if, as you bring R to 0, you also bring the size of the masses to zero.    If you actually pass 2 objects into each other, gravity decreases, because part of mass 1 is now on both sides of part of mass 2, cancelling out some of the gravitation.   Newton's equation no longer applies if the objects are partially inside each other.
Another way to think about this is that, if you were able to build a house in the center of the earth - there would be zero gravity, cause the mass of the earth would be equal all around you - essentially tugging on you from all directions equally.   The same principal would apply if 2 objects occupy the same space.  The gravitational attraction between 2 options in the same space would actually go towards zero, not infinity.   The gravitation would be at it's highest point around the time they touch, or shortly after. - that's an interesting question in fact where it would be highest, but it would be either just when they touch or not far past that.
A: There is no established quantum theory of gravity.
Hence, at the microscopic level of particle, we don't know what is going on gravitationally between particle, but it isn't going to be the "inverse square law" we know, just like electromagnetism between two charged particles is, on quantum scales, not just an "inverse square law", but a rich variety of interactions that can be thought of as being mediated by virtual photons, and that gives the inverse square law of electrostatics only in the macroscopic and non-relativistic limit of the simplest of these interactions.
Essentially, particles don't clump together as you imagine because they are, on the smallest scales, not particles at all. They are quantum states that are smeared out over an area like an electron orbital in an atom, and they do not behave as our classical intuition leads us to believe.
In essence, because we don't have a good description of gravity at these scales, and because quantum objects are not little dots of mass in space, your question might as well be countered with: "Why should they?"
A: The paradox you describe is even worse than the effect of gravity alone: Electrostatics works the same, attracting most matter at everyday distances very strongly by comparison. That is obvious for opposite charges, but even neutral matter attracts as electric interaction induces dipoles. Even where you have equal charges, which do repel each other, you do not end up with a stable floating arrangement (the Earnshaw theorem).
The answer is quantum mechanics: Electrons are fermions, which cannot occupy the same place in phase-space (the Pauli exclusion principle). Hence there are electronic states in atoms which, if occupied, will not be available to other electrons, not even partially (except in special mathematically orthogonal arrangements, the other states). Anything else coming close enough for wavefunctions to overlap experiences a repulsive force, which for any significant overlap is rather large. The result, at least if one were to visualize electrons as point-like rather than occupying their entire orbital, is that matter is essentially empty space.
If you are thinking more on astronomical or cosmological scales, your focus on gravity is correct. In that case, you balance inertia from the big bang with gravity (and possibly other forces, e.g. dark energy). For astronomical bodies, gravitational collapse occurs and is only slowed by conservation of angular momentum, which means that a part of the potential energy must first be turned into heat, which creates gas pressure that balances gravity. Then fusion occurs and that heating balances further collapse. Eventually, the same quantum mechanical effect, the Pauli exclusion principle, stabilizes a neutron star (where the electrons and protons have reacted to neutrons), and if that is not enough, you get a black hole. Which is the other answer to your question: Sometimes nothing does succeed in stopping gravitational collapse. I was temped to throw in the word "ultimately," but note that at least small black holes are not forever, because they shrink as they turn their mass into Hawking radiation.
EDIT:
The accepted and vastly more popular answer essentially points out that we do not know what ultimately happens in detail at the core of a black hole because we do not have a model (or theory) for that. That is certainly true: Even if you acknowledge attempts at building better theories (String theories tend to automatically be theories of quantum gravity), they certainly are not at a state where we could confidently predict what unknown fundamental particles with what masses yet await discovery. And as densities rise, just like electrons and protons react to neutrons to allow further compression, eventually one gets to ever more massive/energetic particles eventually beyond what we could currently have experimental knowledge of. But does it matter just how exotic the particles and energy inside of a black hole looks like in detail? As long as you stay on the outside, you will not get to see much useful evidence of its innermost interior anyways (in theory, there is information in Hawking radiation; in practice, that is useless thermodynamic entropy). Hence the question can largely be answered without a theory of quantum gravity, as this answer attempts to do.
A: I think it helps to look at why things get stuck at all.
If you have two things that could together get to a lower energy level than they are at now, and they can give that excess energy up to something (like sending photons into deep space) that isn't going to give it back (at least for a long time), then they can get stuck (at least for a while).
So why isn't a simple inverse square force law doing that?  
First, if you have a central force law (force between the centers of the bodies), then angular momentum is conserved, so if you don't aim it absolutely perfectly straight one, then it can slingshot around like a planetary orbit rather than getting super close.
Second, it has to be able to give the excess energy away.  If you just have a conservative force, then the potential energy and kinetic energy exchange, which means as you get super close, your kinetic energy gets super large.  And a large kinetic energy means it is hard to get you to not keep going in that general direction.  An extreme case could be an infinite kinetic energy for an instant as you pass through each other, but then you start to move away.  So you got close, but without giving energy away you aren't stuck together, you just pass through each other while going really really fast.
In reality an inverse square law isn't totally accurate at small distances, and the infinite kinetic energy that implied would be a giveaway about that. Quantum effects or relativistic effects can cause deviations from a Newtonian Force Law.
A: First of all, infinitesimal small distances are not allowed in the quantum world simply due to the Heisenberg uncertainty principle - the Newtonian force law doesn't hold at these distances. Apart from that, new forces arise that repel at short distances, a simple $H_2$ molecule is an example for that. When you manage to collide particles instead, the whole realm of particle reactions opens - they don't behave as simple spheres, but react with each other.
