Rare-earth nickelates with the infinite-layer crystal structure have been synthesized in thin film and powder form via topotactic oxygen reduction of the perovskite phase. The infinite-layer phase exhibits remarkable properties, such as superconductivity and magnetic excitations with extraordinarily large bandwidth. Yet, superconductivity was exclusively reported for infinite-layer nickelate films, while polycrystalline powder samples of similar composition were insulating at all measured temperatures. Here, a high-pressure method was used to synthesize high-quality single crystals of the perovskite nickelate La$_{1-x}$Ca$_{x}$NiO$_3$ that were subsequently reduced to the infinite-layer phase La$_{1-x}$Ca$_{x}$NiO$_{2+delta}$. The obtained samples were characterized by X-ray diffraction, electron microscopy, Raman spectroscopy, magnetometry, and electrical transport measurements. Notably, the metal-like electrical conductivity of the infinite-layer crystals is reminiscent of weakly hole-doped infinite-layer thin films. Moreover, local electron energy-loss spectroscopy reveals close similarities between the electronic structures of the crystals and thin films. This work demonstrates the realization of infinite-layer nickelate crystals with macroscopic size as well as superior crystalline quality, and paves the way for future studies exploring whether more heavily Ca-substituted crystals host superconductivity in analogy to sufficiently hole-doped films.