The origin of life is believed to have progressed through an RNA world, in which RNA acted as both genetic material and functional molecules. The structure of the evolutionary fitness landscape of RNA would determine natural selection for the first functional sequences. Fitness landscapes are the subject of much speculation, but their structure is essentially unknown. Here we describe a comprehensive map of a fitness landscape, exploring nearly all of sequence space, for short RNAs surviving selection in vitro. With the exception of a small evolutionary network, we find that fitness peaks are largely isolated from one another, highlighting the importance of historical contingency and indicating that natural selection would be constrained to local exploration in the RNA world.