Let me first ask the question for two-dimensional compact, connected manifolds and orbifolds.
Then, if the answer is *No*, one can remove various conditions on the dimension,
and allow non-compact examples and disconnected examples, to realize a (perhaps) wider range of rationals.

This came up after a class I'm teaching and I didn't know the answer.

Related:

- MO question "Euler characteristic of orbifolds."
- Wikipedia table for 2-dim orbifolds

non connectedexamples it seems to me that the answer isyes. In fact, for any $n$ there is an orbifold whose Euler characteristic is $1/n$, and it is well known that any rational number can be written as a finite sum of fractions of this form (egyptian fraction representation). – Francesco Polizzi Apr 12 '17 at 22:54