Integrated Washington photometry has been obtained for the globular cluster system of NGC 1399, the central elliptical galaxy in the Fornax cluster. A new integrated metallicity index, C-T1, is defined which is about 3 times more sensitive to metal abundance than the previous such Washington index. Abundances accurate to 0.3 dex are derived for ∼150 of the brightest globular clusters. The clusters have a broad, flattened abundance distribution, with a mean metallicity of [Fe/H] = ∼0.85 ± 0.1, 0.75 dex more metal-poor than the halo. A significant fraction of the clusters have an approximately solar abundance, while virtually no clusters exist with [Fe/H] < -2. The luminosity function is also determined for ∼600 clusters to a limiting magnitude of T1∼23.75 (50% completeness limit). The luminosity function clearly shows a turnover at T1 = 23.0 ± 0.15 or V = 23.45 ± 0.16. Matching this luminosity function with that for the globular cluster system of the Galaxy and M87 yields a distance modulus of 31.0 ± 0.25 for Fornax, 0.5 ± 0.2 mag closer than Virgo. A Hubble constant of 66 ± 14 km s-1 Mpc-1 is derived assuming a canonical value for the Local Group Virgocentric infall velocity. Alternatively, solving for this infall velocity gives a value of 236 km s-1.