The existence of classical nonradiating electromagnetic sources is one of the puzzling questions to date. Here, we investigate radiation properties of physical systems composed of a single ultrahigh permittivity dielectric hollow disk excited by electric or magnetic pointlike dipole antennas, placed inside the inner bore. Using analytical and numerical methods, we demonstrate that such systems can support anapole states with total suppression of far-field radiation and thereby exhibit the properties of electric or magnetic nonradiating sources. It is shown that the suppression of the far-field radiated power is a result of the destructive interference between radiative contributions of the pointlike dipole antennas and the corresponding induced dipole moments of the hollow disk. The experimental investigation of the nonradiating electric source has been performed to confirm our theoretical predictions. Our results pave the way to create and realize compact nonradiative sources for applications in modern wireless power transfer systems, sensors, RFID tags, and medical technologies.