The high-pressure equation of state and elastic properties of solid He (He-4) have been calculated using density functional theory formulated in the framework of the exact muffin-tin orbitals method. The theoretical results, obtained within the generalized gradient approximation for the exchange-correlation functional, are in good agreement with the experimental data available for pressures between 13 GPa and 32 GPa. We predict that at 0 K the hexagonal phase of He remains mechanically and thermodynamically stable up to the highest pressure considered in the present study (similar to 150 GPa). The calculated anisotropy ratios of He are similar to those observed in the case of hexagonal metals with c/a similar to 1.63. On the other hand, we find that hydrostatic pressure has negligible effect on the anisotropy of He. This indicates that He can be used as a quasihydrostatic medium in high-pressure experiments up to at least 150 GPa.