Is there any way to define the orientation of an orientable smooth manifold using sheaves (when our smooth manifold is viewed as a locally ringed space) without our definition being overly contrived? 



Orientation of a smooth manifold using sheaves
In the study of (finite-dimensional?) paracompact and locally compact (?) spaces there is Verdier's topological duality theorem, expressed in terms of a dualizing complex (which is built up from a sheafification process using duals of compactly-supported cohomologies of open subspaces, or something like that). It is pure topology, having nothing to do with ringed spaces (just like the orientation sheaf!). In the special case of smooth (paracompact) manifolds, this recovers the orientation sheaf up to a shift on each connected component. It is analogous to the fact that the super-abstract dualizing complexes in Grothendieck duality for (quasi-)coherent cohomology collapses to the old friend "top-degree relative differentials" (up to shifting) in the smooth case. 


But that's all just fancy mumbo-jumbo which puts the orientation sheaf into a broader perspective (like many duality theories for cohomologies). This does not qualify as a good way to initially "define" the orientation sheaf, much as appealing to Grothendieck duality would be a strange (and even circular, from some viewpoints) way to "define" top-degree relative differentials in the smooth case. To get a real theorem out, we have to put some content in.


It seems more illuminating at a basic level to understand how the orientation sheaf is constructed using punctured neighborhoods along the lines of Emerton's comment or the oriented double cover as in David Roberts' answer, and how one can remove some orientability hypotheses in some classical results on "constant coefficient" cohomology by instead allowing coefficients in the locally constant sheaf given by the orientation sheaf. And likewise to understand why the constant sheaf associated to $\mathbf{Z}(1) = \ker(\exp)$ has $n$th tensor power that serves as an orientation sheaf on a complex manifold of dimension $n$ (and so the *natural* isomorphism $\mathbf{C}(1) = \mathbf{C}$ via multiplication explains the absence of needing to choose orientations for various cohomological calculations on complex manifolds (very relevant if one is to have a hope to translating things into algebraic geometry). 


