HR 8799 is an hF0 mA5 γ Doradus-, λ Bootis-, Vega-type star best known for hosting four directly imaged candidate planetary companions. Using the CHARA Array interferometer, we measure HR 8799's limb-darkened angular diameter to be 0.342 ± 0.008 mas (an error of only 2%). By combining our measurement with the star's parallax and photometry from the literature, we greatly improve upon previous estimates of its fundamental parameters, including stellar radius (1.44 ± 0.06 R⊙), effective temperature (7193 ± 87 K, consistent with F0), luminosity (5.05 ± 0.29 L ⊙), and the extent of the habitable zone (HZ; 1.62-3.32 AU). These improved stellar properties permit much more precise comparisons with stellar evolutionary models, from which a mass and age can be determined, once the metallicity of the star is known. Considering the observational properties of other λ Bootis stars and the indirect evidence for youth of HR 8799, we argue that the internal abundance, and what we refer to as the effective abundance, is most likely near solar. Finally, using the Yonsei-Yale evolutionary models with uniformly scaled solar-like abundances, we estimate HR 8799's mass and age considering two possibilities: 1.516+0.038 -0.024 M⊙and 33+7-13.2 Myr if the star is contracting toward the zero-age main sequence or 1.513 +0.023 -0.024 M⊙ and 90+381 -50 Myr if it is expanding from it. This improved estimate of HR 8799's age with realistic uncertainties provides the best constraints to date on the masses of its orbiting companions, and strongly suggests they are indeed planets. They nevertheless all appear to orbit well outside the HZ of this young star.