By utilizing the large multiplexing advantage of the Two-degree Field spectrograph on the Anglo-Australian Telescope, we have been able to obtain a complete spectroscopic sample of all objects in a predefined magnitude range, 16.5 <b_j < 19.7, regardless of morphology, in an area toward the center of the Fornax Cluster of galaxies. Among the unresolved or marginally resolved targets, we have found five objects that are actually at the redshift of the Fornax Cluster; i.e., they are extremely compact dwarf galaxies or extremely large star clusters. All five have absorption-line spectra. With intrinsic sizes of less than 11 HWHM (corresponding to approximately 100 pc at the distance of the cluster), they are more compact and significantly less luminous than other known compact dwarf galaxies, yet much brighter than any globular cluster. In this paper we present new ground-based optical observations of these enigmatic objects. In addition to having extremely high central surface brightnesses, these objects show no evidence of any surrounding low surface brightness envelopes down to much fainter limits than is the case for, e.g., nucleated dwarf elliptical galaxies. Thus, if they are not merely the stripped remains of some other type of galaxy, then they appear to have properties unlike any previously known type of stellar system.