Illustrator does not have the ability to show dimension labels without the cumbersome task of creating them yourself. And even in so doing they would not adjust either their length or dimension number when the measured objects are moved relative to one another.
Illustrator is at it's base a vector drawing program. While it is extremely robust, it does not have every possible feature. Dimensioning is more in the realm of CAD programs and layout programs.
In the past when I have needed exact number spacing between objects or to know the exact spacing between existing objects I have used these different techniques-
One is to simply draw a line (or any shape) at the required spacing length and use it as a temporary spacing device to move your objects to. I usually will give it a magenta stroke so it is easily identifiable and can be deleted or hidden later.
Another way to have specific spacing between objects is to put the objects exactly next to each other and with the Selection Tool (Black Arrow) selected hit Enter. This brings up a dialog box into which specific numbers can be entered and the selected object can be either moved or copied. You can also move selected objects using the transform panel (Window> Transform) by adding or subtracting from the X and Y fields- the input boxes accept simple arithmetic (like xxx.xxx pts+100).
If you need to know the spacing between 2 objects that are other than horizontally or vertically spaced then draw a line between the 2 and look in the Transform Panel under the Line Properties and you can see it's exact length and angle.
Finally, there is a paid plug in for Illustrator called HotDoor CADTools which brings some CAD useability to Illustrator and, of course, the dimensioning you are wanting is included with that.
(I have no affiliation with them)