If a point has no dimension and no area how can there be space? Everyone knows that a point on any plane has zero magnitude, height, width, or volume. A line is a collection of points, and a plane is a collection of lines, while a cube is a collection of planes and so forth.

How is it then that any line, surface or cube can be made of points that have no magnitude area etc.
In order to calculate the "sum" of lengths/areas/volumes etc. for a collection of subsets, you need the collection of subsets to be countable. But if you take a curve or surface etc. and look at all the points on the set, it is uncountable. So you can't reason that the total length/area/volume etc. is $0 + 0 + \ldots = 0$. If anything, you have to consider that e.g. for length of $S$ along the line you get $dx + dx+ \ldots$, "summed" over all points in $S$, which is indeed zero for a countable point set $S$ but for all points on a line segment you get an integral (not countable sum) of $dx$, which gives the length of the line segment.
I think the idea is that lines are sets of points. The length of a line is equal to the cardinality of the set of the points that comprise it. 

Said otherwise, let line segment $P_{1}P_{3}$ designate the set $\langle P_1,P_2,P_3 \rangle$. The line segment is 3 units long. 

In this way, assigning actual measures, such as inches, to units isn't any different than assigning actual things to numbers. For example, $2$ apples. 

Math is abstract. Abstraction involves leaving things out (alternatively, concretion involves adding things in; however you prefer). Leaving out physical things allows us to generalize, which is a reason that math is so broadly applicable.