We present high-resolution aperture synthesis images from the Submillimeter Array of the 225 GHz (1.3 mm) continuum and (CO)-C-12 J = 2-1 line emission from the disks around components of the hierarchical triple system SR 24, located in the Ophiuchus star-forming region. The most widely separated component, SR 24 S (with a projected semimajor axis a = 832 AU), has a circumstellar disk with properties typical of those around single T Tauri stars. The binary SR 24 N (a = 32 AU) is undetected in the continuum but has strong, resolved emission that likely originates in a circumbinary disk with a central gap. The data constrain the total disk mass in the SR 24 N system to be less than or similar to 10(-3) M-., and indicate that the depletion of CO onto dust grains is not more than 100 times larger than the mean value in the interstellar medium. The SR 24 N disk is unusual in that it is only detected in line emission. It is possible that other low-mass disks around binaries and single stars may have been missed in single-dish continuum surveys.