The canonical problem of an electric Hertzian dipole radiating in the presence of a pair of concentric double negative metamaterial spheres is investigated analytically and numerically. The spatial distribution of the near field as well as the total radiated power are examined. The results are compared to those for the corresponding structures made of conventional double positive materials. It is shown that electrically small concentric metamaterial spheres can be designed to be resonant and that these resonant designs lead to significant changes in the field radiated by the electric Hertzian dipole and, in particular, to significant enhancements of the total power radiated by it. The impact of the location of the electric Hertzian dipole in both radiating and scattering configurations is studied in detail. Furthermore, the influence of dispersion and loss is investigated. Finally, a few results for larger size metamaterial concentric spheres are given.