In a formal theory like ZF (or ZFC), one can say that proper classes don't exist, since the formal language can only speak of sets. Inside the formal theory, one can prove sentences stating for instance that there is no universal set. But if we are looking "from the outside" at a model for ZF, we can speak of collections of objects (for instance the collection of all sets) that do not correspond to sets in the model. We call them classes to distinguish the informal from the formal language.
From a technical point of view, the situation is no more paradoxical than for instance a formal theory of the rational numbers in which it can be proved that there is no square root of 2, but that can be embedded in a larger structure where $\sqrt{2}$ does exist.
But it's a bit disturbing if one wants to think of set theory as a foundation of mathematics, describing the real mathematical universe. It seems weird to be able to give specific examples of things that don't exist. With everyday non-existing things like circles with integer radius and perimeter you can't, because there simply aren't any. It reminds me of the great Master Cerebron in Stanislaw Lem's "The cyberiad", who lectured for 47 years on the three types of dragon, each of which doesn't exist, but in completely different ways.
To understand the situation, I think one has to see the problem that ZF was meant to solve. In the late 1800's, Gottlob Frege devised a much simpler formal theory of sets that turned out to be inconsistent due to the Russell paradox of the set of all sets that aren't members of themselves. Restoring Frege's logic turned out to be quite a challenge. Russell suggested a hierarchy of types, while ZF goes in another direction, squeezing set theory into first-order logic. I'm not familiar with other theories, but it seems that whatever we do, Rusell's paradox will keep haunting us one way or another. Finally, since I'm currently reading it, I can't resist mentioning Logicomix.
