The smallest length scale that physicists can think of, is the Planck length, which is roughly 1.6 x 10^{-35} meters or about 10^{-20} times the size of a proton. But physics at such a short length scale is not completely understood yet, although many believe that the string theory is the theory for physics at the Planck length scale.

There is something called the Heisenberg uncertainty principle that prevents anything from getting truly small because its location in space becomes diffuse and undefined. This is a fundamental law of quantum mechanics and a fundamental property of physics: it is difficult to explain in detail without a great deal of college-level math. This "fuzziness" of where something is located in space does not prevent particles from being packed into each-other, but it does make it seem like they form tiny clouds rather than act as single particles.

This said, Einstein's theory of general relativity predicts the existence of singularities - points in space that are infinitely dense and where time literally ends.

Black holes contain such singularities as the theory goes. Obviously, this disagrees with the Heisenberg uncertainty principle, so one or the other (or both!) of the two theories must fail when you look at the interior of black holes, but nobody knows which or how the theories break when it does happen.

Matter can get quite small! The smallest known particles are subatomic particles such as quarks, which comprise protons and neutrons, and electrons. These particles are described in terms of their masses, in units of meVc^-2. Electrons have a mass of about 0.5 meVc^-2, which corresponds to roughly 10^-30kg (1 in the 30th decimal place!). To put this into perspective, an average adult human weighs around 70kg! I hope this helps!

Everything is made of atoms, and that's the smallest thing that has the different chemical properties of different things. Oxygen atoms, nitrogen atoms, carbon atoms, for example. If an oxygen atom, for example, is broken apart, it isn't oxygen any more. There are electrons and protons and neutrons, but those aren't oxygen, they're just the particles that make up all matter. Those things can break down into smaller particles that high energy physicists study.

Atoms are between 30 and 300 millionths of a meter across - a meter is just a bit bigger than a yard. Protons and neutrons are smaller than atoms, of course, and electrons are about 1000 times smaller than protons or neutrons.