We report extensive radial velocity observations of this unusual K-type "Vega-excess" star and show that each component of the well-known close visual pair is itself a spectroscopic binary. The primary is a single-lined binary with a period of 262 days, and the companion is a double-lined binary with a 315 day period. The difference in the center-of-mass velocities of HD 98800 A and B combined with astrometric information allow us to place tight constraints on the orbit of the wide pair. We show that if the pair is bound, the stars must now be approaching perisystem in an extremely eccentric, long-period orbit which does not leave enough room for a circumstellar dust disk around either of the spectroscopic binaries. However, there is no proof yet that the two visual components are physically connected, and it seems rather unlikely that we happen to be observing a system like this so close to maximum approach. We discuss alternative scenarios that seem just as unlikely and conclude that the stability of the disk is not yet understood.