Paradox are all 4D distances zero? The 4 dimension distance from the origin of a point is $\sqrt{x^2+y^2+z^2-t^2}$. Which means the 4 dimensional distance on the light-cone is zero.
Take a point A and a point B in the future at roughly the same position. Now move away from point A at the speed of light (and so going no 4D distance.) And then move back towards B at the speed of light. The total 4D distance is zero.
If you have gone two distances which are 0 then surely the total distance travelled is zero?
But yet the 4D distance from A to B is not zero.
If a geodesic is the shortest distance between two points in 4D space. Then moving along rays of light would surely be the smallest?
What have I done wrong in this paradox?
 A: Geodesics are not necessarily the shortest path between two points. In fact timelike geodesics are always the longest timelike curve connecting two points (or more specifically a local maximum of the proper time). Spacelike geodesics on the other hand are (only) saddle points of the path length.
A: Your "paradox" is not specifically related to 4D Minkowski space. (Also, your confusion seems to be related to the Minkowski metric, not the dimensionality of spacetime, as the "paradox" already works in 1+1 dimensions.)
Basically, you show that two spacetime points at, say, $x_A=(t_A,0,0,0)$ and $x_B=(t_B,0,0,0)$ can be linked by a purely lightlike path of Minkoswki length zero (e.g. by using several zig-zag components). On the other hand, you can also joint it by a timelike path of squared length $-(t_A-t_B)^2$. Of course, you also can choose a path composed out of purely spacelike segments with a positive squared length (zig-zags outside the lightcone). What is the "distance of the points"?
You can do essentially the same in simple 2D Euclidean space: There are (ininfinitely) many paths joining two given points, with different lengths. Here the distance is intuitively obvious: It is the shortest possible length among all those paths (it's the length of a straight line between the points, and all other paths are longer "detours"). 
In Minkowski space, you use the same argument, you just have to take the "squared length", since you cannot in general take the square root: The squared distance is the smallest possible squared length. (Note that there is some possible confusion here regarding the signs of the metric.) Those paths are again straight lines, or "geodesics", and the statement is that geodesics "maximise eigentime" (which includes anothe minus sign).
