ECH (embedded contact homology) capacities give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. We compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called 'concave toric domains'. Examples include the (nondisjoint) union of two ellipsoids in R-4. We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. For example: (1) we calculate the Gromov width of every concave toric domain; (2) we show that many inclusions of an ellipsoid into the union of an ellipsoid and a cylinder are 'optimal'; and (3) we find a sharp obstruction to ball packings into certain unions of an ellipsoid and a cylinder.
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