We study equilibrium configurations of thin and elongated non-Euclidean elastic strips with hyperbolic two-dimensional reference metrics ? which are invariant along the strip. In the vanishing thickness limit energy minima are obtained by minimizing the integral of the mean curvature squared among all isometric embeddings of ?. For narrow strips these minima are very close to minimal surfaces regardless of the specific form of the metric. We study the properties of these "almost minimal" surfaces and find a rich range of three-dimensional stable configurations. We provide some explicit solutions as well as a framework for the incorporation of additional forces and constraints.