Is it possible to do a radius constraint in Blender? Is it possible to do a radius constraint in Blender? 
For example, consider 2 cubes, 1 unit apart, as the following;

[]--1 unit--[]


If we move the box on the right up or down, it will retain same distance as the following;

    []
    /
   1 unit 
  /
[] 


This is not for Blender Game Engine, I simply want to set up constraints, but can't see of how to get it done in Blender.
I think what you are looking for is the Limit Distance constraint.

From the wiki:


  The Limit Distance constraint forces its owner to stay either further from, nearer to, or exactly at a given distance from its target. In other words, the owner’s location is constrained either outside, inside, or at the surface of a sphere centered on its target.