Is it possible to build a 3D model of the genetic code so that the axes are GUAC (each in some order), and each amino acid is contiguous?

No.

Can prove this by considering L, R, S together. See http://www.hgvs.org/mutnomen/codon.html. No ordering of ACU rows that alllow all three to be next to each other.

The best you can do is have one of L, R, or S have two pieces.