(Abridged) Aims: We obtained VLT/FLAMES+UVES high-resolution, fibre-fed spectroscopy (FFS) of five young massive clusters in M83 (NGC 5236). This forms the basis of a pilot study testing the feasibility of using FFS to measure the velocity dispersions of several clusters simultaneously, in order to determine their dynamical masses; Methods: We adopted two methods for determining the velocity dispersion of the star clusters: cross-correlating the cluster spectrum with the template spectra and minimising a chi^2 value between the cluster spectrum and the broadened template spectra. Cluster 805 in M83 was chosen as a control to test the reliability of the method, through a comparison with the results obtained from a standard echelle VLT/UVES spectrum obtained by Larsen & Richtler; Results: We find no dependence of the velocity dispersions measured for a cluster on the choice of red giant versus red supergiant templates, nor on the method adopted. We measure a velocity dispersion of sigma_los = 10.2+/-1.1 km/s for cluster 805 from our FFS. Our FLAMES+UVES velocity dispersion measurement gives M_vir = (6.6+/-1.7)e5 M_sun, consistent with previous results. This is a factor of ~3 greater than the cluster's photometric mass, indicating a lack of virial equilibrium. However, based on its effective star formation efficiency, the cluster is likely to virialise, and may survive for a Hubble time, in the absence of external disruptive forces; Conclusions: We find that reliable velocity dispersions can be determined from FFS. The advantages of observing several clusters simultaneously outweighs the difficulty of accurate galaxy background subtraction, providing that the targets are chosen to provide sufficient S/N ratios, and are much brighter than the galaxy background. Comment: 10 pages, 4 figures, accepted by A&A