Spectroscopic observations of binary stars in globular clusters are essential to shed light on the poorly constrained period, eccentricity, and mass ratio distributions and to develop an understanding of the formation of peculiar stellar objects. 47 Tuc (NGC 104) is one of the most massive Galactic globular clusters, with a large population of blue stragglers and with many predicted but as-yet elusive stellar-mass black holes. This makes it an exciting candidate for binary searches. We present a multi-epoch spectroscopic survey of 47 Tuc with the VLT/MUSE integral field spectrograph to determine radial velocity variations for 21699 stars. We find a total binary fraction in the cluster of (2.4+/-1.0)%, consistent with previous photometric estimates, and an increased binary fraction among blue straggler stars, approximately three times higher than the cluster average. We find very few binaries with periods below three days, and none with massive dark companions. A comparison with predictions from state-of-the-art models shows that the absence of such short-period binaries and of binaries with massive companions is surprising, highlighting the need to improve our understanding of stellar and dynamical evolution in binary systems.