A mesoscopic system of a few particles can undergo changes of configuration that resemble phase transitions but with a nonuniversal behavior. A notable example is orientational melting, in which localized particles with long-range repulsive interactions forming a two-dimensional crystal become delocalized in common closed trajectories. Here we report the observation of orientational melting occurring in a two-dimensional crystal of up to 15 ions. We measure density-density correlations to quantitatively characterize the occurrence of melting, and use a Monte Carlo simulation to extract the angular kinetic energy of the ions. By adding a pinning impurity, we demonstrate the nonuniversality of orientational melting and create novel configurations in which localized and delocalized particles coexist. Our system realizes an experimental testbed for studying changes of configurations in two-dimensional mesoscopic systems, and our results pave the way for the study of quantum phenomena in ensembles of delocalized ions.