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](https://mathoverflow.net/q/51993/6094)."
* [Wikipedia table](https://en.wikipedia.org/wiki/Orbifold#2-dimensional_orbifolds) for 2-dim orbifolds



Is every rational realized as the Euler characteristic of some manifold or orbifold?
Products of 2-orbifolds with manifolds will do the trick. There are 2-orbifolds of Euler characteristic $1/n$ (take a quotient of $S^2$ by a rotation of order $2n$). Then take a product with a manifold of Euler characteristic $m \in \mathbb{Z}$ to get all rationals. 


