Using computer simulations, we study the diffusion, interactions, and strain fields of point defects in a face-centered-cubic crystal of hard spheres. We show that the vacancy diffusion decreases rapidly as the density is increased, while the interstitial diffusion exhibits a much weaker density-dependence. Additionally, we predict the free-energy barriers associated with vacancy hopping and find that the increasing height of the free-energy barrier is solely responsible for the slowing down of vacancy diffusion. Moreover, we find that the shape of the barriers is independent of the density. The interactions between vacancies are shown to be weakly attractive and short-ranged, while the interactions between interstitials are found to be strongly attractive and are felt over long distances. As such, we find that vacancies do not form vacancy clusters, while interstitials do form long-lived interstitial clusters. Considering the strain field of vacancies and interstitials, we argue that vacancies will hardly feel each other, as they do not substantially perturb the crystal, and as such exhibit weak interactions. Two interstitials, on the other hand, interact with each other over long distances and start to interact (attractively) when their strain fields start to overlap.