The Carina Nebula is one of the major massive star-forming regions in the Galaxy. Its relatively nearby distance (2.35kpc) makes it an ideal laboratory for the study of massive star formation, structure and evolution both for individual stars and stellar systems. Thanks to the high-quality spectra provided by the Gaia-ESO survey and the LiLiMaRlin library, as well as Gaia EDR3 astrometry, a detailed and homogeneous spectroscopic characterization of its massive stellar content can be carried out. Our main objective is to spectroscopically characterize all massive members of the Carina Nebula in the Gaia-ESO survey footprint to provide an updated census of massive stars in the region and an updated estimate of the binary fraction of O stars. We performed accurate spectral classification using an interactive code that compares spectra with spectral libraries of OB standard stars, as well as line-based classic methods. We calculated membership using our own algorithm based on Gaia EDR3 astrometry. To check the correlation between the spectroscopic n-qualifier and the rotational velocity, we used a semi automated tool for the line-broadening characterization of OB stars based on a combined Fourier transform and goodness-of-fit methodology. The Gaia-ESO survey sample of massive OB stars in the Carina Nebula consists of 234 stars. The addition of brighter sources from the Galactic O-Star Spectroscopic Survey and additional sources from the literature allows us to create the most complete census of massive OB stars done so far in the region. It contains a total of 316 stars, with 18 of them in the background and 4 in the foreground. Of the 294 stellar systems in Car OB1, 74 are of O type, 214 are of non-supergiant B type and 6 are of WR or non-O supergiant (II to Ia) spectral class. We identify 20 spectroscopic binary systems with an O-star primary, of which 6 are reported for the first time, and another 18 with a B-star primary, of which 13 are new detections. The average observed double-lined binary fraction of O-type stars in the surveyed region is 0.35, which represents a lower limit. We find a good correlation between the spectroscopic n-qualifier and the projected rotational velocity of the stars. The fraction of candidate runaways among the stars with and without the n-qualifier is 4.4% and 2.4%, respectively, although non resolved double-lined binaries could be contaminating the sample of fast rotators.