Single spins in solid-state systems are often considered prime candidates for the storage of quantum information, and their interaction with the environment the main limiting factor for the realization of such schemes. The lifetime of an excited spin state is a sensitive measure of this interaction, but extending the spatial resolution of spin relaxation measurements to the atomic scale has been a challenge. We show how a scanning tunneling microscope can measure electron spin relaxation times of individual atoms adsorbed on a surface using an all-electronic pump-probe measurement scheme. The spin relaxation times of individual Fe-Cu dimers were found to vary between 50 and 250 nanoseconds. Our method can in principle be generalized to monitor the temporal evolution of other dynamical systems.