Within a few years, we hope to be able to see the imprint of the Universe’s first stars on their surrounding hydrogen, using observations of the 21-cm hyperfine line. This line is redshifted into the low-frequency radio, where it could be detected by a sensitive radiometer in lunar orbit. A concept for an experiment to do so (‘DARE’) has been developed, and its ground-based prototypes are undergoing development and testing. Detailed circuit modeling of these instruments has allowed us to build a calibration and signal extraction pipeline based on a Markov Chain Monte Carlo (MCMC) technique. This pipeline must also contend with signals from the Galaxy, Sun and Moon which are orders of magnitude more powerful than the cosmological signal. From the ground it must, in addition, deal with contamination of the spectrum by the Earth’s ionosphere, and distortions to the spectrum introduced by having to deal with a complex environment and anthropogenic radio interference. We demonstrate the advantages of a lunar orbit in allowing us to extract the cosmological signal by eliminating some of these spurious effects, and show the level of precision to which the properties of the instrument must be known. A fully self-consistent, rigorous treatment of errors on the spectrum, allowed by the MCMC code, allows us to show how well an experiment such as DARE can distinguish between different physical models of the Cosmic Dawn.