We have identified 82 short-period variable stars in Sextans A from deep Wide Field Planetary Camera 2 observations. All the periodic variables appear to be short-period Cepheids, with periods as small as 0.8 days for fundamental mode Cepheids and 0.5 days for first-overtone Cepheids. These objects have been used, along with measurements of the red giant branch (RGB) tip and red clump, to measure a true distance modulus to Sextans A of μ0 = 25.61 ± 0.07, corresponding to a distance of d = 1.32 ± 0.04 Mpc. Comparing distances calculated by these techniques, we find that short-period Cepheids (P < 2 days) are accurate distance indicators for objects at or below the metallicity of the Small Magellanic Cloud. As these objects are quite numerous in low-metallicity star-forming galaxies, they have the potential for providing extremely precise distances throughout the Local Group. We have also compared the relative distances produced by other distance indicators. We conclude that calibrations of RR Lyrae stars, the RGB tip, and the red clump are self-consistent, but that there appears to be a small dependence of long-period Cepheid distances on metallicity. Finally, we present relative distances of Sextans A, Leo A, IC 1613, and the Magellanic Clouds.