 Second homology of a closed nonorientable three-manifold

Zero Euler characteristic and $H_3=0$ give free rank $b_1-1$. The universal coefficient theorem for cohomology identifies $H^3(M;\mathbb{Z})=\mathbb{Z}/2$ with $\mathrm{Ext}(H_2(M;\mathbb{Z}),\mathbb{Z})$, so the only torsion in $H_2$ is one copy of $\mathbb{Z}/2$. In particular $b_1\ge 1$.