Geophysical fluids such as the ocean and atmosphere can be stratified: their density depends on the depth. As a consequence, they can host internal gravity waves that propagate in the bulk of the fluid and transport energy and momentum far from the surface. These waves can be carried over large distances, thereby influencing large-scale circulation patterns and climate. In the ocean, the interplay between heat diffusion and salt diffusion can lead to the formation of spatially periodic density profiles called thermohaline staircases. Drawing an analogy from electronic band structure of one-dimensional crystals, we show with laboratory experiments the existence of band gaps, ranges of frequencies over which the wave propagation is prohibited in the presence of a periodic stratification. We also demonstrate the existence of surface states arising at the boundary of the resulting internal wave crystal, and elucidate their relation with topological insulators. Our results suggest that periodic stratifications can profoundly affect internal wave dynamics in geophysical fluids ranging from the Arctic Ocean to giant planet interiors.