Functor category's objects fail to be a class? Given two categories, we can form the functor category whose objects are functors. Functors by definition consist of two mappings from in general classes to classes, which makes it fail to be a set. But set theory in which classes are involved avoids Rusell's paradox by only allowing sets as members. This argument shows that functor category's objects fail to be a class, which makes functor category fail to be a category. 
I've never been trained in axiomatic set theory, so I guess my first question should be:is there anything wrong with the above argument? If it works, then how can we overcome the difficulty?
There are other similar problems bothering me: in say K-theory, we take the equivalence classes of modules/vector bundles as our objects. However the equivalence class of any module is always a proper class, just as we can't talk about the set consisting of all sets with a single element. This is a somehow "universal" problem, since equivalence classes appear everywhere, and we're technically almost never allowed to use the construction whenever classes are involved. 
 A: This can be circumvented (to some degree) if we use Grothendieck universes instead of classes. Roughly speaking, a set $\mathcal U$ is a universe if it's closed under all the operations we use in ZF. A set is called $\mathcal U$-small if it's [isomorphic to] an element of $\mathcal U$, and we add an axiom that every set is contained in a universe. Then we fix a $\mathcal U_0$ and work only with $\mathcal U_0$-small sets. Then if we run into a set $X$ that is no longer $\mathcal U_0$-small, we simply take a universe $\mathcal U_1$ containing $\mathcal U_0\cup\{X\}$ and work with $\mathcal U_1$-small sets.
Of course, this method will not work if you're interested in some general construction which necessarily keeps knocking you out of your universe. See free completion as a pseudomonad on nLab for an example.
Disclaimer: I am not trained in set theory either
A: As has been discussed, there are plenty of ways to set up the foundations and definitions so that it is not a problem to have proper classes of proper classes, rather than of sets.
But you are right to worry about functor "categories".  In Mac Lane's language, a small category has a set of objects and a set of morphisms.  A locally small category has a class of objects, and between any two objects a set of morphisms — this is what most commonly is given the name "category".  A large category may have a class of morphisms between some pair of objects.  The point is that the functor category between any two small categories is again small, and the functor category from a small category to a locally small one is locally small.  But the functor category between two locally small categories might be large.  Consider for example the category whose objects are sets and whose morphisms are isomorphisms; then in its endofunctor category identity functor has $\prod_c c! = 2^{\sum_c c}$ automorphisms, where $c$ ranges over cardinalities and $c!$ is the cardinality of the set of automorphisms of a set of size $c$ (we have $2^c \leq c! \leq c^c$, but $c^c \leq (2^c)^c = 2^{c^2} = 2^c$ for infinite $c$).
