The category $\operatorname{Sh}_X^{Set}$ of $\operatorname{Set}$ valued sheaves is equivalent to the category $Et/X$ via the adjunction given by the section functor $\Gamma$ and $\Delta:F\to\prod_{x\in X}F_x=Et(F)$ where $Et(F)$ has the finest topology which makes all the sections of $F$ continues.
I'd say that in general the local section of such an étale space doesn't have a (say) group structure, nor the stalks/fibers, then the section sheaf is just a $\operatorname{Set}$ valued sheaf. So, in general, do I have an étale space associated to a sheaf which take values in any abelian category?
EDITED
In effect, if $F$ is a presheaf of abelian groups each stalk has a group stucture, which is inherithed by the étale space and the (local) sections. So I'd rather ask the following:
1) How does it work with a general abelian category? (but it may be worthless to think about sheaves via étale space in such a general context)
2) Which is the equivalent category $\operatorname{Sh}^{Ab}_X\cong{}?\subset Et/X$ given by the adjunction above?