The title is the question. I should perhaps add that $\mathbb R$ has its usual topology and the product has the product topology, i.e., $\mathbb R^{\mathbb R}$ is the space of all functions from $\mathbb R$ to $\mathbb R$ with the topology of pointwise convergence. It is easy to see that this space is separable, e.g., the set of polynomials with rational coefficients is dense. However, in general separable topological spaces, subspaces need not be again separable.
I guess that the answer to this question is well-known -- however searching, e.g., Engelking's book for such a special question is rather frustrating.