We show that packed, horizontally aligned films of single-walled carbon nanotubes are hyperbolic metamaterials with ultrasubwavelength unit cells and dynamic tunability. Using Mueller matrix ellipsometry, we characterize the films' optical properties, which are doping level dependent, and find a broadband hyperbolic region tunable in the mid-infrared. To characterize the dispersion of in-plane hyperbolic plasmon modes, we etch the nanotube films into nanoribbons with differing widths and orientations relative to the nanotube axis, and we observe that the hyperbolic modes support strong light localization. An agreement between the experiments and theoretical models using the ellipsometry data indicates that the packed carbon nanotubes support bulk anisotropic responses at the nanoscale. Self-assembled films of carbon nanotubes are well-suited for applications in thermal emission and photodetection, and they serve as model systems for studying light-matter interactions in the deep subwavelength regime.