Dispersionless energy bands in k space are a peculiar property gathering increasing attention for the emergence of novel electronic, magnetic, and photonic properties. Here, we explore the impact of electronic flat bands on the light-matter interaction. The van der Waals interaction between the atomic layers of hexagonal boron nitride induces flat bands along specific lines of the Brillouin zone. The macroscopic degeneracy along these lines leads to van Hove singularities with divergent joint density of states, resulting in outstanding optical properties of the excitonic states. For the direct exciton, we report a giant oscillator strength with a longitudinal-transverse splitting of 420 meV, a record value, confirmed by our ab initio calculations. For the fundamental indirect exciton, flat bands result in phonon-assisted processes of exceptional efficiency, that compete with direct absorption in reflectivity, and that make the internal quantum efficiency close to values typical of direct band gap semiconductors.