For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf.

Is there some sense, or some class of examples (that include true orbifolds, namely orbifolds that are not schemes), in which these numbers have some enumerative meaning?

One class of examples is Gorenstein orbifolds, namely those orbifolds whose all degree shifting numbers are integers. In this case one can use the crepant resolution conjecture to translate the question (at least in genus $0$) into a question of enumerativity of Gromov-Witten invariants for the crepant resolution.

But what about other cases? In particular, can one make some enumerative sense of the Gromov-Witten invariants that are obtained when at least one of the cohomology classes is chosen to be the cohomology class of a twisted sector?

(apologies if the question is too vague, and if it is a repetition of other similar questions. I am not sure if this should be community wiki.)