Compact groups (CGs) of galaxies, similar to those catalogued by Hickson, appear to be the densest galaxy structures in the Universe. Redshift information is insufficient to determine whether a CG is roughly as dense in 3D as it appears in projection, or whether it is caused by a chance alignment (CA) along the line of sight within a larger galaxy system. Recent precise distance measurements help probe the nature of the nearest CG, situated in the Virgo cluster, whose dominant member is M60. The isolated status of the CG is reassessed with recent photometry and a statistical analysis is performed on the surface brightness fluctuation (SBF) distances measured by Mei et al. in Virgo, for 4 of the 5 CG members. The neighboring galaxy NGC 4606 appears (with 80-90% confidence) to be too faint to affect the isolated status of the CG. Taken at face value, the SBF distances suggest that M59 and NGC 4660 lie roughly 2 Mpc to the foreground of M60, NGC 4638, and the bulk of the Virgo cluster. The statistical analysis confirms that the CG is, indeed, the result of a CA of its galaxies, with NGC 4638 lying at least 800 kpc further away (with 99% confidence) than either M59 or NGC 4660. The first two galaxy distances are consistent with M59 and NGC 4660 constituting a tight pair. The dominant galaxy, M60, is at least 440 kpc more distant (95% confidence) than the M59+NGC 4660 pair, and over 1 Mpc (99% confidence) more distant if one uses the broken linear calibration of the SBF distances. This work constitutes the first direct analysis of the nature of a compact group of galaxies. CAs of galaxies represent a realistic alternative to truly dense groups to explain the nature of CGs. With current SBF distance accuracies, one could determine the nature of HCG 68 in the same way.