We utilise multi-epoch MUSE spectroscopy to study binary stars in the core of the Galactic globular cluster NGC 3201. Our sample consists 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 of observations. We introduce a statistical method to determine the probability of a star showing radial velocity variations based on the whole inhomogeneous radial velocity sample. Using HST photometry and an advanced dynamical MOCCA simulation of this specific cluster we overcome observational biases that previous spectroscopic studies had to deal with. This allows us to infer a binary frequency in the MUSE field of view and enables us to deduce the underlying true binary frequency of (6.75 ± 0.72)% in NGC 3201. The comparison of the MUSE observations with the MOCCA simulation suggests a large portion of primordial binaries. We can also confirm a radial increase in the binary fraction towards the cluster centre due to mass segregation. We discovered that in the core of 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 blue straggler stars and the connection to SX Phoenicis-type stars. We show evidence that two blue straggler formation scenarios, the mass transfer in binary (or triple) star systems and the coalescence due to binary-binary interactions, are present in our data. We also describe the binary and spectroscopic properties of four sub-subgiant (or red straggler) stars. Furthermore, we discovered two new black hole candidates with minimum masses (M sin i) of (7.68 ± 0.50) M_⊙, (4.4 ± 2.8) M_⊙, and refine the minimum mass estimate on the already published black hole to (4.53 ± 0.21) M_⊙. These black holes are consistent with an extensive black hole subsystem hosted by NGC 3201.