As Martin Brandenburg said, cannot exist a (locally euclidean) smooth variety structure, but in the more large category $(C^\infty-Ring)^{op}$ I think that the product exist (see "Models for Smooth Infinitesimal Analysis" Ieke Moerdijk, Gonzalo E. Reyes, see T. 2.8 pag. 30).

It is a funtor $P: C^\infty\to Set$ that map $\mathbb{R}^n$ on the set of funtion like
$f\circ\pi_J:\prod_i\mathbb{R}_i \to \prod_J\mathbb{R}_j\to \mathbb{R}^n $ where $J\subset I$ is a finite subset, $\pi_J$ is the natural projection, and $f$ a smooth map (and by composition on morphisms). We have to show that $P$ is the sum of $I$ copies of $\mathbb{R}$ (where $\mathbb{R}$ is see as a $C^\infty$ spaces naturally). Give a manifold M, a morphism of $C^\infty$-spaces $\mathbb{R}\to M$ is uniquely represented by a smooth map $M\to \mathbb{R}$. Then give morphisms $g'_i: \mathbb{R}\to M$ i.e. smooth maps $g_i: M\to \mathbb{R}$ i.e. a map $g=\prod_i g_i: M\to \prod_I \mathbb{R}$ follow the morphism $g': P\to M$ such that on argument $g'_n: P(\mathbb{R}^n)\to M(\mathbb{R}^n)$ map $f\circ \pi_J $ (where $J=${$j_1,...,j_n$}) on $(f\circ g_{j_1}, ..., f\circ g_{j_n}): M\to \mathbb{R}^n$.

(I hope this work...)