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Here is a different example. Let $S$ be an Enriques surface. Then $X = S \times \mathbb{CP}^1$ is uniruled, hence has Kodaira dimension $-\infty$. But $b_{1}(X) = 0$, hence $h^{0,1}(X)=0$. Then we can can conclude that $X$ is not rational or birational to any Fano since for example $\pi_{1}(X) = \mathbb{Z}_{2}$.