A presheaf can be defined on any category: it is just another name for a contravariant functor from that category to any other category. In particular, if you have a topological space $X$ you can associate to it a category $\mathcal Open (X)$ whose objects are the open subsets $U\subset X$ and the morphisms the inclusions $U\subset V$. A presheaf in the usual sense is then indeed a functor from $\mathcal Open (X)$ to another category. Now you have a full subcategory $\mathcal Pathconopen (X)\subset \mathcal Open (X)$ obtained by taking only path connected open subsets of $X$ and their inclusions, and you can call a conravariant functor from it a presheaf . All fine and well. However it is impossible to say when a presheaf is a sheaf in that context: the problem is that if a path connected open set $U\subset X$ is covered by path connected open subsets $U_i$, the intersections $U_i\cap U_j$ will in general not be path connected and the sheaf conditions thus don't make sense.