We use the word paradox in a couple different ways. One is in logical contradictions, i. e. "This sentence is false." It is impossible to have the statement be either true or false, so it is a paradox. "A married bachelor" is another paradoxical statement, as you are definitionally bound by the word "bachelor" to be unmarried.

Another way we use paradoxes is in seemingly contradictory statement that do actually have an answer. Zeno's paradox falls in this category. Like you say, there is the definite answer that Achilles does catch the tortoise, the seemingly contradictory part is the description that you have to cover an infinite number of ever smaller but never 0 distances to go anywhere, so how can you move?

The answer being that the time it takes to move those smaller increments also decreases to 0 at exactly the same rate.