Short answer: You can't take sheaves of topological spaces. This is because a "sheaf of abelian groups" is in fact an abelian group object in the category of sheaves. This equivalence of concepts does not hold for topological spaces because the forgetful functor adjunction of Top and Set is not monadic. If you read Mac Lane's book Sheaves in Geometry and Logic, they explain precisely what this means and why it's important.

Long answer (I am not fully competent to answer this part): taking sheaves of topological spaces can be done, but the enrichment must be in the category of compactly generated (weak hausdorff) spaces or else we won't have some things that we want like being cartesian closed.