Today a friend asked me if the Moebius strip with one segment identified to a point is orientable or not. The first thing I replied is that it is not a manifold so you can't define orientability in terms of volume forms.
It made me wonder if the notion of orientability can be generalized to spaces with slightly less restrictive conditions than manifolds (like the space proposed by my friend). Spaces that look locally like $\mathbb{R}^n$ in almost every point but are not a manifold (at least not with the induced topology).
Is there any useful generalization of orientability for spaces similar to manifolds but not so well behaved?
P.S. Is there a general notion of orientability, e.g. for the rationals? I saw this question but I'm precisely interested in the last point of his question, i.e. a notion generalizing the concept of orientability for manifolds for similar spaces with worse conditions.