Is there a nontrivial Riemannian metric for which a geodesic equation doesn't have any solutions? What would such a metric mean physically?
No, it does not exist. Given a Riemannian manifold with metric of class $C^2$, for every initial point and every initial vector, there is a unique maximal geodesic satisfying those initial conditions. That is a straightforward consequence of the existence and uniqueness theorem for integral curves on manifolds (here the tangent space manifold).
For the existence $C^1$ is enough, but it does not guarantee uniqueness. With less regularity it is not possible to write down the geodesic equation. (Actually one could require that the metric is differentiable but the derivatives are not continuous, in this case nor the existence is guaranteed. On the other hand such metric has continuous derivatives on a dense set.)