smc

Let $r$ be the radius, let $C$ be the center of the cube, and let $P$ be the center of one of the eight outer spheres, noting that $PC=2r$. Back, in the corner of the unit cube, a smaller cube whose inner corner coincides with $P$. The cube is of dimensions $1\times 1\times 1$ and its space diagonal is of length $\sqrt{3}$. Thus the space diagonal of the unit cube is $2(r\sqrt{3}+2r)=\sqrt{3}$. Solving for $r$, we get $\boxed{\frac{2\sqrt{3}-3}{2}}$