We introduce hydra (hyperbolic distance recovery and approximation), a new method for embedding network- or distance-based data into hyperbolic space. We show mathematically that hydra satisfies a certain optimality guarantee: it minimizes the 'hyperbolic strain' between original and embedded data points. Moreover, it is able to recover points exactly, when they are contained in a low-dimensional hyperbolic subspace of the feature space. Testing on real network data we show that the embedding quality of hydra is competitive with existing hyperbolic embedding methods, but achieved at substantially shorter computation time. An extended method, termed hydra+, typically outperforms existing methods in both computation time and embedding quality.