I will just repeat the title:



> 
> Is there a closed non-smoothable 4-manifold with zero Euler
>  characteristic?
> 
> 
> 


I am guessing yes simply based on other existence theorems I have seen for 4-manifolds.



Is there a closed non-smoothable 4-manifold with zero Euler characteristic?
The Kirby-Siebenmann invariant in $H^4(M;\Bbb Z/2)$, an obstruction to smoothability, is additive under connected sum in dimension 4. In even dimensions, $\chi(M \# N) = \chi(M) + \chi(N) -2$. 


To construct manifolds with nontrivial Kirby-Siebenmann invariant we should apply Freedman's theorem: simply connected topological 4-manifolds are determined by their intersection form and Kirby-Siebenmann invariant; if the intersection form is odd, both Kirby-Siebenmann invariants are realized, and if the intersection form is even, only one KS-invariant is realized. So there is a topological manifold $F(\Bbb{CP}^2)$, homotopy equivalent to $\Bbb{CP}^2$ but with nontrivial Kirby-Siebenmann invariant.


So for the desired non-smoothable manifold with zero Euler characteristic, one can take (for instance) $$F(\Bbb{CP}^2) \# 3\Bbb{CP}^2 \#(S^2 \times \Sigma\_2).$$


