I've been searching around and haven't found a satisfying answer to this. A lot of sources just give examples of supposedly reversible processes (such as pool balls hitting each other) actually losing very small amounts of heat to friction, drag, or some other usually ignored facet of the system.
Looking for more rigorous definitions to try and figure it out, I went to Wikipedia. But this only made everything seem circularly defined: reversibility is defined with the condition of no change in entropy, which is a state defined using the Carnot Cycle, which assumes reversibility. I assume this is merely a fault of Wikipedia's nature; the articles can't all coordinate and avoid circularity.
Thinking about it on my own, I couldn't reason why a possible (yet astronomically improbable) process cannot be truly reversible. For instance, take just two identical particles in a vacuum with equal initial velocities away from each other. Of course, this will likely never happen, but it's at least theoretically possible. Gravity should pull them back to their initial positions. Where is the energy transfered or converted here?
In essence, what exactly is reversibility, and how do we know it can't happen in a finite time?