Context. The Gaia-ESO Survey (GES) is a large public spectroscopic survey at the European Southern Observatory Very Large Telescope. Aims: A key aim is to provide precise radial velocities (RVs) and projected equatorial velocities (vsini) for representative samples of Galactic stars, which will complement information obtained by the Gaia astrometry satellite. Methods: We present an analysis to empirically quantify the size and distribution of uncertainties in RV and vsini using spectra from repeated exposures of the same stars. Results: We show that the uncertainties vary as simple scaling functions of signal-to-noise ratio (S/N) and vsini, that the uncertainties become larger with increasing photospheric temperature, but that the dependence on stellar gravity, metallicity and age is weak. The underlying uncertainty distributions have extended tails that are better represented by Student's t-distributions than by normal distributions. Conclusions: Parametrised results are provided, which enable estimates of the RV precision for almost all GES measurements, and estimates of the vsini precision for stars in young clusters, as a function of S/N, vsini and stellar temperature. The precision of individual high S/N GES RV measurements is 0.22-0.26 km s-1, dependent on instrumental configuration. Based on observations collected with the FLAMES spectrograph at VLT/UT2 telescope (Paranal Observatory, ESO, Chile), for the Gaia- ESO Large Public Survey (188.B-3002).Full Table 2 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/580/A75