Imagine there's a pure, disembodied intellect, and you somehow have the ability to communicate with it. It's a very clever intellect, so it's perfectly receptive to abstract, a priori mathematics: but it has never had any experience of spatial things, and it wants you to explain space to it. How might you go about this? Well, first you might explain the number line. You invite it to consider an infinite set of objects (we'll call them 'real numbers'), all different from one another, but continuously ordered in two directions from a particular element that we'll call 'zero' (or 'the origin'): ever greater to a positive infinity, and ever less to a negative infinity. And now, with the number line in place, you invite the intellect to take three such lines. That is to say, you invite it to consider an infinity of ordered triples of the form <x, y, z>, where x, y and z are all real numbers from this same set, but are capable of varying independently of one another. Let's call each of these triples a ...

I've nothing against Sean Greenberg's answer, but I figured I'd just add a word or two on a further relevant distinction here. Infinite divisibility is not the same as the possibility of dividing something into infinitely many parts. At least, it doesn't need to be understood in that way. There's a distinction that goes back at least as far as Aristotle, between the actual infinite and potential infinite, and the notion of infinite divisibility can be interpreted in either way. If we interpret infinite divisibility in the sense of the potential infinite (which, for what it's worth, is how Aristotle himself understood -- and endorsed -- the concept), this will mean that, no matter how small something might be, it can still be divided into still smaller parts. You can cut something into two halves, divide each of those to yield four quarters, divide each of these to yield eight eighths, and carry on going without ever needing to stop dividing. Mathematically, there is no greatest power of two: so, no...

I shall begin with a 'philosophical' kind of answer, the kind of answer that philosophers ever since Aristotle's time might have given. (Indeed, it is closely related to the answers that Aristotle himself gave to Zeno's paradoxes of motion. Perhaps you're already familiar with those paradoxes: but, if not, then I'd invite you to look them up, for you might enjoy pondering them). I think the flaw in your question lies in that phrase "the next moment". In the case of space, you seem to be treating it as continuous in the sense that, between any two points, no matter how close they might be, there will still be further spatial points between them -- so that to jump straight from one to the other would have to involve some sort of teleportation, bypassing all those intervening points. And yet (as a philosopher might tell you) time itself is equally continuous, and in exactly the same way. At any given moment of time, there is simply no such thing as the next moment. The continuous nature of time...