1. The problem asks to find the volume of a cube with side length $\frac{4}{9}$ miles. 2. The formula for the volume $V$ of a cube with side length $s$ is: $$V = s^3$$ 3. Substitute the side length $s = \frac{4}{9}$ into the formula: $$V = \left(\frac{4}{9}\right)^3$$ 4. Calculate the cube of the fraction: $$V = \frac{4^3}{9^3} = \frac{64}{729}$$ 5. The volume of the cube is $\frac{64}{729}$ cubic miles. This fraction is already in simplest form, so the final answer is: $$\boxed{\frac{64}{729}}$$ cubic miles.