Osmotic properties of dilute salt-free solutions of stiff-chain polyelectrolytes are studied using computer simulations. A number of factors affecting the charge distribution around the macroions are analyzed quantitatively for parameters of an aqueous solution of a typical synthetic strongly charged rodlike polyelectrolyte. Departing from a mean-field treatment of an infinitely long linear charged rod, we significantly refine the model description. We found that incorporation of effects such as electrostatic correlations between counterions, chain flexibility, and addition of salt decreases the osmotic coefficient slightly. On the other hand, a model of a finite macroion with a noncentral charge distribution predicted significantly higher pressure. We quantify all these contributions separately and show what deviation one should expect from the prediction of the Poisson-Boltzmann solution of the infinite rod cell model. We further comment on the discrepancy between our simulation results and experimentally measured values.