The empty set can be regarded as an object in the category of smooth manifolds, at least for technical considerations.
Is the empty set an orientable manifold?
EDIT: This is wrong. See Henning Makholm's answer below.
An $n$-manifold is orientable if and only if it has a nonzero differential form taking $n$ arguments. The empty manifold has no nonzero forms whatsoever, so it is not orientable in any dimension.
Contradicting Espen's answer: The empty map does (vacuously) provide a nonzero differential form at every point on the manifold, and does so continuously. Therefore the empty manifold is orientable in every dimension.
(But really this will depend on the exact definition you're working with, and among the various usually-assumed-equivalent definitions for "orientable" there are some that are only really equivalent when the manifold is assumed to be nonempty).
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