The static structure of salt-free solutions of rod-like polyelectrolyte solutions is studied using Monte Carlo simulations and integral equations. The polymer reference interaction site model theory with the reference Laria-Wu-Chandler closure is found to be in good agreement with the simulations for the polyion-polyion and counterion-counterion correlation functions, and in qualitative agreement with the simulations for the counterion-polyion correlation functions. The effect on the polyion concentration on the counterion distribution is found to be important at low concentrations and becomes more significant when the Bjerrum length or degree of polymerization are increased. The theory is in almost quantitative agreement with the simulations for the static structure factor. The theory predicts that all three partial structure factors display low angle peaks at roughly the same wave vectors. The position and intensity of the peak in the polyion structure factor is insensitive to the Bjerrum length. These predictions are consistent with recent experiments.