There can never be a truly empty space. That was the opinion of many scholars from the times of ancient Greece up to the beginning of the twentieth century. When the idea of aether as a medium in which light can travel has been refuted, the existence of vacuum became widely accepted. But then the quantum revolution came, and nothing is ever simple with quantum physics.
The main obstacle in achieving space that is entirely empty is the Heisenberg uncertainty principle. It states that the position and momentum of an object can never be known exactly. This is, furthermore, not just due to technical imperfections in measuring these quantities; the object itself does not know them exactly.
Let us now take a glass cell and pump all air out. If we also leave it in complete darkness, there will be no light and, therefore, no electromagnetic field and no atoms or molecules inside, right?
Not quite. Light is an oscillating electromagnetic field and as such can be mathematically described as a harmonic oscillator, similar to a pendulum. And a harmonic oscillator has a position and momentum which, even at ground state (i.e., with no light), cannot be exactly zero but have some uncertainty. So there still is some electromagnetic field present, even in complete darkness!
But things can get even weirder because in quantum physics, virtually everything can be described as a harmonic oscillator. For every kind of particles, there can be defined a field whose excitations are the respective particles. For light, there is the electromagnetic field and the particles are photons, electrons are excitations in an electron field, and so on. And each harmonic oscillator has to follow the uncertainty principle. In our glass cell, we thus have a small bit of fluctuations of the electromagnetic field but also fluctuations for electrons and other particles. Vacuum is an endlessly boiling soup where every now and then an electron pops out and disappears again, then a quark, then something else.
Does all that sound ridiculous? It turns out that these phenomena have observable effects. Take, for instance, two metallic plates placed in vacuum. One would naively expect that nothing will happen to them since they are in vacuum. But we know better — there are always fluctuations, and these will be smaller in the space between the plates than everywhere around. As a result, the plates will attract each other; in a different configuration than parallel, they could even repel. This behaviour is known as Casimir effect (though I am stretching things a bit here — only the fluctuations of the electromagnetic field are important for the Casimir effect) and has already been observed in an experiment.
Another, even more important evidence of fluctuations of the vacuum is the existence of spontaneous emission. If you excite an atom (for example by shining light on it) it will eventually radiate the energy it absorbed and end up in its ground state. But from the point of view of classical physics, this happens only when there is electromagnetic field around the atom. This means that an excited atom in utter darkness should stay excited — but it does not! This can only be explained by quantum physics; fluctuations in the vacuum are strong enough to kick the atom to its ground state while emitting a photon, similarly to the presence of electromagnetic field in the classical picture.
So remember — vacuum (for instance the vast empty space between you and the nearest star when watching the skies at night) is not empty. It is alive with many particles that we can never directly see, swirling around. And nature maybe, after all, really is scared of emptiness.