Observations of binary stars in globular clusters are essential to shed light on the formation of peculiar stellar objects, such as blue straggler stars, and to probe the populations of dark stellar remnants thought to be located in the dense cluster cores. 47 Tuc is one of the most massive galactic globular clusters and its large population of blue stragglers and a predicted but as yet elusive population of up to hundreds of black holes make it an exciting candidate for binary searches. Using multi-epoch data from the VLT/MUSE integral field spectrograph, we analyse the binary population of 47 Tuc and identify 708 binary candidates based on the variability of their radial velocities. We present a new estimate of the total binary fraction in the cluster and find a significantly enhanced binary fraction among blue straggler stars, suggesting an origin involving multiple star systems. The determination of orbital parameters and minimum companion masses allows us to put tight constraints on the number of black holes in binary systems in 47 Tuc. By confronting the observations with state-of-the-art model predictions, we reveal a surprising absence of short-period binaries with massive companions. This highlights the need to improve our understanding of stellar evolution in binary systems.