  In this paper, we compute the embedded contact homology (ECH) capacities of
the disk cotangent bundles $D^*S^2$ and $D^*\mathbb{R} P^2$. We also find sharp
symplectic embeddings into these domains. In particular, we compute their
Gromov widths. In order to do that, we explicitly calculate the ECH chain
complexes of $S^*S^2$ and $S^* \mathbb{R} P^2$ using a direct limit argument on
the action inspired by Bourgeois's Morse-Bott approach and ideas from
Nelson-Weiler's work on the ECH of prequantization bundles. Moreover, we use
integrable systems techniques to find explicit symplectic embeddings. In
particular, we prove that the disk cotangent bundles of a hemisphere and of a
punctured sphere are symplectomorphic to an open ball and a symplectic bidisk,
respectively.
