Magnetic and dielectric resonances in the sub-terahertz (sub-THz) frequency range are observed in pure and Al-substituted hexagonal barium ferrite. A resonator based on magnetic excitations has been fabricated and its performance characteristics have been studied. The possible use of the resonator at sub-THz frequencies has been demonstrated. The resonator exhibited a loaded Q-factor of 150-330 in the frequency range 97-108 GHz. Dielectric resonances in a single-crystal barium hexaferrite are observed in the frequency range 75-110 GHz. The modes excited by circularly polarized electromagnetic waves show nonreciprocal propagation characteristics. The dielectric resonances may occur at a much higher frequency than ferromagnetic resonance. It is shown that degeneracy in the dielectric modes is lifted with an applied magnetic field H and that the modes can be tuned by 10 GHz or more with H. Data on frequencies of the modes versus H shows hysteresis. Theoretical predictions on H-tuning characteristics of the principal dielectric E11δ mode are in agreement with the data. The dielectric modes are of importance for the realization of low-loss devices, including resonators, isolators and phase shifters.