Embedded Polyhedron

Relevant wiki: Centroid of a Triangle

Here's a view looking straight at a vertex of the outer octahedron.

The cube touches the outer triangles at their centers which are 1/3 of the way in.

The sides of the inner octahedron are thus 1/3 as long as the outer one.

The outer octahedron has $3^{3}=\boxed{27}$ times the volume.

The radius of a circumscribed sphere of a octahedron with sides $a$ is $r_{oc} = \frac{\sqrt{2}}{2}a$ and the radius of a inscribed sphere of a octahedron with sides $a$ is $r_{oi} = \frac{\sqrt{6}}{6}a$ (see here ).

The radius of a circumscribed sphere of a cube with sides $a$ is $r_{cc} = \frac{\sqrt{3}}{2}a$ and the radius of a inscribed sphere of a cube with sides $a$ is $r_{ci} = \frac{1}{2}a$ (see here ).

If the smallest octahedron has a side $r$ , then the bigger octahedron has a side $\frac{\sqrt{2}}{2} \cdot \frac{2}{1} \cdot \frac{\sqrt{3}}{2} \cdot \frac{6}{\sqrt{6}}r = 3r$ . This makes the ratio of the sides of the bigger octahedron to the smaller octahedron $\frac{3r}{r} = 3$ , so the ratio of volumes is $3^3 = \boxed{27}$ .