We study the eigenstate properties of a nonintegrable spin chain that was recently realized experimentally in a Rydberg-atom quantum simulator. In the experiment, long-lived coherent many-body oscillations were observed only when the system was initialized in a particular product state. This pronounced coherence has been attributed to the presence of special "scarred" eigenstates with nearly equally spaced energies and putative nonergodic properties despite their finite energy density. In this paper we uncover a surprising connection between these scarred eigenstates and low-lying quasiparticle excitations of the spin chain. In particular, we show that these eigenstates can be accurately captured by a set of variational states containing a macroscopic number of magnons with momentum pi. This leads to an interpretation of the scarred eigenstates as finite-energy-density condensates of weakly interacting pi magnons. One natural consequence of this interpretation is that the scarred eigenstates possess long-range connected correlations in both space and time. We verify numerically the presence of this spatiotemporal long-range order and explain how it is consistent with established no-go theorems precluding its existence in ground states and at thermal equilibrium.