There is a generalization of a presheaf called a "fibered category" or a "grothendieck fibration".  This is analogous to the etale space construction for presheaves on O(X).  Every presheaf in the sense of a presheaf taking values in Sets (most other constructions come from enriching presheaves of sets) can be identified with a very simple type of fibered category.  In general, fibered categories wth a fixed cleavage (something like a skeleton of pullbacks) define a contravariant pseudofunctor taking values in the 2-category of categories.  It is only a pseudofunctor because composition is not in general strictly associative, merely associative up to unique isomorphism.  You should check out Vistoli's book on descent, fibered categories, and grothendieck topologies here: http://homepage.sns.it/vistoli/descent.pdf .  

But to answer your question, to generalize the etale space for sheaves, you'll have to introduce the idea of descent for 1-stacks, then sheaves become degenerate stacks, i.e. 0-stacks.  If you're trying to deal with sheaves without resorting to too much category theory, you have to remember that presheaves of abelian groups, for example, are presheaves of abelian group objects in Sets.  All of the categories that you've mentioned are monadic with respect to the forgetful functor adjunction with Sets, so we can always just take objects of that sort in the category of sets.  It's the reason why a "presheaf of topological spaces" doesn't really make a lot of sense, since Top is not algebraic over Sets.  So there is no good way to define sheaves taking values in an arbitrary category without substantially increasing the generality.

If you aren't familiar with what I'm talking about specifically, Mac Lane's "sheaves in geometry and logic" has a very detailed explanation of how the etale space works, and how we can produce sheaves that take values in algebraic categories, but not arbitrary categories.  It also has a very in-depth construction of the etale space for presheaves of sets and also proves the equivalence of categories between the full subcategory of sheaves and the full subcategory of "spaces" called etale spaces.