Memristors, memcapacitors, and meminductors, collectively called memelements, represent an innovative generation of circuit elements whose properties depend on the state and history of the system. The hysteretic behavior of one of their constituent variables, under the effect of an external time-dependent perturbation, is their distinctive fingerprint. In turn, this feature endows them with the ability to both store and process information on the same physical location, a property that is expected to benefit many applications ranging from unconventional computing to adaptive electronics to robotics, to name just a few. For all these types of applications, it is important to find appropriate memelements that combine a wide range of memory states (multi-state memory), long memory retention times, and protection against unavoidable noise. Although several physical systems belong to the general class of memelements, few of them combine all of these important physical features in a single component. Her we demonstrate theoretically a superconducting memory structure based on solitonic long Josephson junctions (LJJs). We show that the Josephson critical current of the junction behaves hysteretically as an external magnetic field is properly swept. According to the hysteretic path displayed by the critical current, a LJJ can be used as a multi-state memory, with a controllable number of available states. In addition, since solitons are at the core of its operation, this system provides an intrinsic topological protection against external perturbations. Solitonic Josephson-based memelements may find applications as memories, and in other emerging areas such as memcomputing, i.e., computing directly in/by the memory. ; Comment: 14 pages, 9 figures