We consider a general relativistic model of a self-interacting complex scalar field with logarithmic nonlinearity motivated by studies of laboratory superfluids and Bose-Einstein condensates. Spherically-symmetric gravitational equilibria are shown in this model, which do not have event horizons but which are regular, singularity-free and asymptotically flat. They can be thus interpreted as compact stars whose stability against gravitational collapse is enhanced not only by the Heisenberg uncertainty principle but also by the property of superfluidity itself, their ``darkness'' comes naturally as a result of suppressed dissipative excitations. Such objects do not obey any absolute upper mass limit of a Tolman-Oppenheimer-Volkoff type, while their relativisticity and effective compactness values are comparable to those of black holes. Their spatial density distribution drops abruptly (at the Gaussian-like rate), which can be mistaken in realistic astronomical observations for the presence of an exact material surface. We therefore present logarithmic superfluid stars as dark compact objects and black hole mimickers.