Results of a Monte Carlo dynamics simulation on the three-dimensional randomly-diluted Ising antiferromagnet Fe0.25Zn0.75F2 indicate that its spin-glass phase at zero magnetic field is characterized by the presence of finite anti ferromagnetic domains, separated by random vacancies, and strongly correlated in space and time. In a microscopic description, the glassy behavior is a consequence of the short range of the interactions in the highly diluted regime. By studying the aging of the system, we also find that its dynamics is in agreement with experimental results and predictions of stochastic short-range models of spin glasses. In particular, the data collapse of the autocorrelation function evidences a power-law time decay with the scaling variable t/t(w), where t(w) is the waiting time before the measurements, as predicted by the droplet model of spin glasses close to the percolation concentration.