Following the recent outburst of the recurrent nova RS Oph on 2006 February 12, we measured its near-infrared size using the IOTA, Keck, and PTI Interferometers at multiple epochs. The characteristic size of ~3 mas hardly changed over the first 60 days of the outburst, ruling out currently popular models whereby the near-infrared emission arises from hot gas in the expanding shock. The emission was also found to be significantly asymmetric, evidenced by nonzero closure phases detected by IOTA. The physical interpretation of these data depends strongly on the adopted distance to RS Oph. Our data can be interpreted as the first direct detection of the underlying RS Oph binary, lending support to the recent "reborn red giant" models of Hachisu & Kato. However, this result hinges on an RS Oph distance of ≲540 pc, in strong disagreement with the widely adopted distance of ~1.6 kpc. At the farther distance, our observations imply instead the existence of a nonexpanding, dense, and ionized circumbinary gaseous disk or reservoir responsible for the bulk of the near-infrared emission. Longer baseline infrared interferometry is uniquely suited to distinguish between these models and to ultimately determine the distance, binary orbit, and component masses for RS Oph, one of the closest known (candidate) Type 1a supernova progenitor systems.