Context: Luminous blue compact galaxies, common at z ~ 1 but now relatively rare, show disturbed kinematics in emission lines. Aims: As part of a programme to understand their formation and evolution, we have investigated the stellar dynamics of a number of nearby objects in this class. Methods: We obtained long-slit spectra with VLT/FORS2 in the spectral region covering the near-infrared calcium triplet. In this paper we focus on the well-known luminous blue compact galaxy ESO 338-IG04 (Tololo 1924-416). A previous investigation, using Fabry-Perot interferometry, showed that this galaxy has a chaotic Hα velocity field, indicating that either the galaxy is not in dynamical equilibrium or that Hα does not trace the gravitational potential due to feedback from star formation. Results: Along the apparent major axis, the stellar and ionised gas velocities for the most part follow each other. The chaotic velocity field must therefore be a sign that the young stellar population in ESO 338-IG04 is not in dynamical equilibrium. The most likely explanation, which is also supported by its morphology, is that the galaxy has experienced a merger and that this has triggered the current starburst. Summarising the results of our programme so far, we note that emission-line velocity fields are not always reliable tracers of stellar motions, and go on to assess the implications for kinematic studies of similar galaxies at intermediate redshift. Based on observations collected at the European Southern Observatory, Paranal, Chile, under observing programme 65.N-0668.