The philosophers of the Eleatic school, analyzing the nature of the movement, came to this paradox: in order for the body to move, it needs emptiness. But what is emptiness? This is what exists, but has no properties and doesn't manifest itself in any way. If emptiness exists, then it can not be known, because it is nothing and doesn't contain any being in itself. Hence, emptiness is something existing and at the same time, not existing. Thus, the assumption of the existence of motion leads to a logical contradiction. Where is the error in this reasoning?
The paradox comes about because there are several statements that are not true. Lets start by examining each, one at a time:
1- In order to move, a body needs emptiness (void, space); true.
2- Emptiness (void, space) is what "exists," that has no properties and doesn't manifest itself in any way; false, (it has length, we can see and feel emptiness (void,space).
3- If emptiness exists, then it can not be known; false, (it can be seen and felt).
4- Because it is nothing; false, it is something, it contains (itself, void, space).
5- Hence, emptiness is something existing, and at the same time not existing; false, second part of statement is false.
Thus, the assumption that a body needs emptiness (void,space) to move, does not lead to a paradox.
An emptiness clearly has a size. The degree to which something can move obviously depends upon the size of the emptiness into which it will be moving.
So it is not devoid of properties, and has some aspects of being. Thus it is not an example of being and not being at the same time.
If emptiness exists, then it can not be known, because it is nothing and doesn't contain any being in itself.
It is possible to define an empty set and know that it has no content.
I agree with jobermark you cannot have a definition of emptiness without a definition of fullness and there is no definition without it. Everything else is a degree of these extremes. Cannot imagine movement in either extremety only the degrees in between.
Let me take a different tack than other respondents.
What is emptiness? It is the metaphorical space in a container.
In fact, motion through space is understood by a conceptual metaphor called Containment which is a hint at how neurons organize themselves at the neurocomputational level. In essence, metaphysical presuppositions that lead to these contradictions you state (and Zeno gave at length), are actually properties of how the brain constructs experience. What does it mean for something to "exist"? Did the atom "exist" before the mind conceived it? These require you to adequately disambiguate whether you are referring to an "atom" or an atom, the former a representation within our heads and the latter to das Ding an sich.
The questions you are asking are answered in different ways depending on your metaphysics which might be naive realism, transcendental idealism, or philosophical realism. As you begin to piece together a collection of propositions that satisfy your biases, you'll have a theory, and if that theory is both rational and empirical, you'll graduate to analytical philosopher!