B

To find the moment of inertia of a hollow cubical box about an axis passing through the centers of two opposite faces, we start with the formula:

\[ I = x \cdot M \cdot a^2 \]
where \( x \) is the constant we want to find, \( M \) is the mass, and \( a \) is the side length of the cube.

The moment of inertia for a hollow cube about that axis is known to be:

\[ I = \frac{1}{3} M a^2 \]

We equate the two expressions:

\[ x \cdot M \cdot a^2 = \frac{1}{3} M a^2 \]

Dividing both sides by \( M \cdot a^2 \) (assuming they are non-zero), we get:

\[ x = \frac{1}{3} \]

Thus, the value of \( x \) is \( \frac{1}{3} \), which corresponds to option B.