Let $x$ and $y$ be real numbers satisfying
\[\frac{2}{x} = \frac{y}{3} = \frac{x}{y}.\]Determine $x^3.$
If I multiply the first and third fractions by $xy$, I get $2y = x^2$. If I multiply the first and second fractions by $3x$, I get $6 = xy$. I can substitute $y = 6/x$ from the second equation into the first equation, and get $2(6/x) = x^2$. Simplifying, I get $12 = x^3$, so $x^3 = 12$. 