We present high-resolution far-UV spectroscopy of the 14 galaxies of the Ly alpha Reference Sample; a sample of strongly star-forming galaxies at low redshifts (0.028 < z < 0.18). We compare the derived properties to global properties derived from multi-band imaging and 21 cm H I interferometry and single-dish observations, as well as archival optical SDSS spectra. Besides the Ly alpha line, the spectra contain a number of metal absorption features allowing us to probe the kinematics of the neutral ISM and evaluate the optical depth and and covering fraction of the neutral medium as a function of line of sight velocity. Furthermore, we show how this, in combination with the precise determination of systemic velocity and good Ly alpha spectra, can be used to distinguish a model in which separate clumps together fully cover the background source, from the "picket fence" model named by Heckman et al. We find that no one single effect dominates in governing Ly alpha radiative transfer and escape. Ly alpha escape in our sample coincides with a maximum velocity-binned covering fraction of less than or similar to 0.9 and bulk outflow velocities of greater than or similar to 50 km s(-1), although a number of galaxies show these characteristics and yet little or no Ly alpha escape. We find that Ly alpha peak velocities, where available, are not consistent with a strong backscattered component, but rather with a simpler model of an intrinsic emission line overlaid by a blueshifted absorption profile from the outflowing wind. Finally, we find a strong anticorrelation between H alpha equivalent width and maximum velocity-binned covering factor, and propose a heuristic explanatory model.