The microscopic structure of fluids of simple spheres is well known. However, the constituents of most real-life fluids are non-spherical, leading to a coupling between the rotational and translational degrees of freedom. The structure of simple dense fluids of spheroids - ellipsoids of revolution - was only recently determined by direct experimental techniques [A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield, and E. Sloutskin, Phys. Rev. Lett. 107, 238301 (2011)]. Using confocal microscopy, it was demonstrated that the structure of these simple fluids cannot be described by hard particle models based on the widely used Percus-Yevick approximation. In this paper, we describe a new protocol for determining the shape of the experimental spheroids, which allows us to expand our previous microscopy measurements of these fluids. To avoid the approximations in the theoretical approach, we have also used molecular dynamics simulations to reproduce the experimental radial distribution functions g(r) and estimate the contribution of charge effects to the interactions. Accounting for these charge effects within the Percus-Yevick framework leads to similar agreement with the experiment.