Metals are canonical plasmonic media at infrared and optical wavelengths allowing one to guide and manipulate light at sub-diffractional length scales. A special form of optical waveguiding is offered by highly anisotropic crystals revealing different signs of the dielectric function along orthogonal directions. These latter types of media are classified as hyperbolic and many crystalline insulators, semiconductors and artificial metal-based metamaterials belong to that class. Layered anisotropic metals are also anticipated to support hyperbolic waveguiding. Yet this behavior remains elusive primarily because interband processes introduce extreme losses and arrest light propagation. Here, we report on the observation of propagating hyperbolic waves in a prototypical layered nodal-line semimetal ZrSiSe. The unique electronic structure with touching energy bands at nodal points/lines suppresses losses and enables a hyperbolic regime at the telecommunications frequencies. The observed waveguiding in metallic ZrSiSe is a product of polaritonic hybridization between near-infrared light and long-lived nodal-line plasmons. By mapping the energy-momentum dispersion of the nodal-line hyperbolic modes in ZrSiSe we inquired into the role of additional screening associated with the surface states.