1. **State the problem:** We need to find the length of side $x$ in a triangle with angles $30^\circ$, $60^\circ$, and the side opposite $60^\circ$ is $\sqrt{7}$. 2. **Identify the triangle type and formula:** This is a $30^\circ$-$60^\circ$-$90^\circ$ right triangle. The sides are in the ratio $1 : \sqrt{3} : 2$ opposite $30^\circ$, $60^\circ$, and $90^\circ$ respectively. 3. **Assign known values:** The side opposite $60^\circ$ is $x_{60} = \sqrt{7}$. According to the ratio, $x_{60} = x_{30} \times \sqrt{3}$. 4. **Find $x_{30}$:** $$ x_{30} = \frac{x_{60}}{\sqrt{3}} = \frac{\sqrt{7}}{\sqrt{3}} $$ 5. **Rationalize the denominator:** $$ x_{30} = \frac{\sqrt{7}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{21}}{3} $$ 6. **Conclusion:** The length of side $x$ opposite $30^\circ$ is $\boxed{\frac{\sqrt{21}}{3}}$ in simplest radical form with a rational denominator.