We utilize multi-epoch MUSE spectroscopy to study binary stars in the core of the Galactic globular cluster NGC 3201. Our sampleconsists of 3553 stars with 54 883 spectra in total comprising 3200 main-sequence stars up to 4 magnitudes below the turn-off.Each star in our sample has between 3 and 63 (with a median of 14) reliable radial velocity measurements within five years ofobservations. We introduce a statistical method to determine the probability of a star showing radial velocity variations based onthe whole inhomogeneous radial velocity sample. Using HST photometry and an advanced dynamical MOCCA simulation of thisspecific cluster we overcome observational biases that previous spectroscopic studies had to deal with. This allows us to infer a binaryfrequency in the MUSE FoV and enables us to deduce the underlying true binary frequency of (6.75±0.72) % in NGC 3201. Thecomparison of the MUSE observations with the MOCCA simulation suggests a significant fraction of primordial binaries. We canalso confirm a radial increase of the binary fraction towards the cluster centre due to mass segregation. We discovered that in the coreof NGC 3201 at least (57.5±7.9) % of blue straggler stars are in a binary system. For the first time in a study of globular clusters,we were able to fit Keplerian orbits to a significant sample of 95 binaries. We present the binary system properties of eleven bluestraggler stars and the connection to SX Phoenicis-type stars. We show evidence that two blue straggler formation scenarios, themass transfer in binary (or triple) star systems and the coalescence due to binary-binary interactions, are present in our data. We alsodescribe the binary and spectroscopic properties of four sub-subgiant (or red straggler) stars. Furthermore, we discovered two newblack hole candidates with minimum masses (Msini) of (7.68±0.50) M, (4.4±2.8) M, and refine the minimum mass estimate onthe already published black hole to (4.53±0.21) M. These black holes are consistent with an extensive black hole subsystem hostedby NGC 3201.