When in history did the notion of space, geometric space appear? I. e. when in history geometric space was treated or thought of as a whole, as the site in which all geometric objects exist? When I think of Euclid's Elements, I have the impression that it just treats relations between segments, areas; straight lines, planes (?), but doesn't treat space as a whole, as the site in which all geometric objects exist. (I am assuming the whole content in Euclid's Elements is what I said it is, but I have never read it thoroughly.)
In general, Euclid's definitions talk about points and lines having a relationship to a plane but he does not say they need a plane in order to live. The only line he says needs a plane is the circle.
I think he had a geometric conception of space as a whole, but it differs from the modern conception of space where a plane or surface are needed for points or lines to live. The latter are called manifolds, but Euclidean geometry, when understood properly, is not a flat manifold because it is not a manifold.