This gives a pretty good explanation… and includes the wise observation that “Any paradox can be treated by abandoning enough of its crucial assumptions.”
Head down to the paragraph that starts “For Zeno’s paradoxes, the most important features of any linear continuum are that”. With that and the descriptions of continuum and limits. I think the article puts it across well.

I think the smallest limit “where an object either takes or doesn’t and takes the one space next to it” would be a Planck length.

@dabbler has got it. In math, you can cut numbers in half on and on forever. In the physical world, you can’t do that. There is a minimum – the Planck length.

Zeno’s paradoxes are mathematical in nature, and their resolution also came from mathematics, not physics, with development of the modern concept of limits by Weirstrauss & others. This allows infinite series to have finite sums without logical paradoxes.

The concept of space being quantized by the Planck length remains an unproven hypothesis not accessible to experimental physics today. People speak of it like it’s a done deal. I don’t think it has much relevance to Zeno’s paradoxes.

As others have said Zeno’s Paradox is mathematical in nature but movement in the physical universe is itself puzzling. What happens when something moves from one place to another? It seems the simplest thing but it may not be fully understood until we can reconcile General Relativity with quantum mechanics.

Zeno’s Paradox reminds me of Euclid’s fifth postulate which was mathematical curiosity for 2,000 years as no one could prove it true.

From my limited understanding of quantum mechanics, it seems to provide a non-intuitive answer to Zeno. It gets around the paradox by allowing particles to move incrementally from A to B without covering anything beteeen A and B. Does this suggest that space and time are also discrete quantities?