According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions (see solvitur ambulando). To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.
Why is this argument insufficient? It would seem to be a prime example of a counterexample proof. If Zeno's argument is that there exists a finite distance journey that cannot be completed due to requiring infinite steps, then I can see that a single counterexample is insufficient, but Zeno seems to be claiming that completion is impossible for all finite journeys. If there exists at least one finite journey that can (somehow) be completed in finite time, then it is impossible that all finite journeys are impossible and so Zeno's argument crumbles.
Analogously, if I claim that at least one black cat exists, showing me an orange cat does nothing. However, if, like Zeno, I claim that all cats are black, proof that there exists one orange cat in a box somewhere completely annihilates my argument.
To be clear, I'm not asking whether all finite distance journeys of the type mentioned by Zeno are impossible. I believe that they are possible. I'm asking what is wrong with Diogenes's argument.
I do know that in many models of physics, escaping the event horizon of a black hole is considered impossible as it would require exceeding the speed of light. Quantum physics and the Planck length are potentially relevant to a quantum-based solution, but clearly Zeno was not considering "move this electron to a point halfway between these two orbitals" as a plausible journey. I am not really talking about these advanced physics scenarios, but the "walk across the room" or "cha-cha across the dance floor" scenarios that Zeno was discussing. Alternately, "cat walks to food bowl" or "chicken crosses the road" could be acceptable scenarios that we can not only envision, but actually perform experimentally to verify that the creature is able to complete the journey before the heat death of the universe.