The highly redshifted 21-cm transition of neutral Hydrogen provides a unique observational method for probing the Dark Ages and Cosmic Dawn eras of the early universe. Due to the extremely wide beams of low frequency experiments, the measured spectrum will naturally be influenced by the surrounding terrain, complicating the separation of the 21-cm signal from systematics, namely the bright foreground emission. We present a discussion of how the angular horizon will alter the foreground spectrum and the required measurement tolerance on the horizon to adequately extract the 21-cm signal. We find that even at locations that appear to be largely “flat,” such as the Murchison Radio-astronomy Observatory in Western Australia (the location of the EDGES experiment), the “lost” horizon can introduce large errors when modeling the beam-weighted foreground, overwhelming the 21-cm spectrum, if not properly accounted for. We also discuss the implications for lunar-based experiments such as the Dark Ages Polarimeter PathfindER (DAPPER), which will possess a horizon that is much simpler to model as the lunar surface is free of vegetation and other intervening obstructions.