I can make a pretty good case for sphere packing of a finite graph in ***some*** $\mathbb R^n.$ From http://en.wikipedia.org/wiki/Circle_packing_theorem#Generalizations_of_the_circle_packing_theorem 
we learn that a nonplanar graph still induces a circle packing on a compact orientable surface of larger genus, the surface having constant curvature. For example, for planar graphs we could realize them either as circle packings in the plane, then simply make those into spheres (thereby done), or we could make a circle packing on $\mathbb S^2$ and then ask whether we can blow up those circles into spheres with the same tangency relationships. The answer is yes, for each circle, take the sphere that intersects $\mathbb S^2$ orthogonally in precisely that circle.

For torus graphs, I am betting on 3-spheres in $\mathbb R^4,$ where we can take the flat Clifford torus. For larger genus, from Nash embedding we can take our compact surface with constant curvature $-1$ in some   $\mathbb R^n.$

Well, somebody just posted an answer, it says so at the top of the page, maybe they actually know something.  



> The circle packing theorem generalizes
> to graphs that are not planar. If G is
> a graph that can be embedded on a
> surface S, then there is a constant
> curvature Riemannian metric d on S and
> a circle packing on (S, d) whose
> contacts graph is isomorphic to G. If
> S is closed (compact and without
> boundary) and G is a triangulation of
> S, then (S, d) and the packing are
> unique up to conformal equivalence. If
> S is the sphere, then this equivalence
> is up to Möbius transformations; if it
> is a torus, then the equivalence is up
> to scaling by a constant and
> isometries, while if S has genus at
> least 2, then the equivalence is up to
> isometries.




EDIT: I suspect it is worth trying to disprove $\mathbb R^3$ for, say, $K_7,$ which is a torus graph. See *Topological Graph Theory* by Jonathan L. Gross and Thomas W. Tucker. If $K_7$ works in  $\mathbb R^3$ try $K_8$ and $K_9,$ the dog graph.

EDIT TOO: some anecdotal evidence, we can always place $K_n$ as the regular simplex in $\mathbb R^{n-1}.$ So now it is a question of how to selectively erase adges in order to get the graph we are actually given...Note that the articles on ball-touching in $\mathbb R^3$ all seem to be about balls of fixed unit radius.