The Hamming dimension of a graph G is introduced as the largest dimension of a Hamming graph into which G embeds as an irredundant induced subgraph. An upper bound is proved for the Hamming dimension of Sierpinski graphs S"k^n, k>=3. The Hamming dimension of S"3^n grows as 3^n^-^3. Several explicit embeddings are constructed along the way, in particular into products of generalized Sierpinski triangle graphs. The canonical isometric representation of Sierpinski graphs is also explicitly described.