Let the sides of the cuboid be $a$, $b$, $c$. The volume is $abc$, and the area of the surfaces are $ab$, $ac$, $bc$. Now if we multiply all the areas and take the square root we get $\sqrt{a^2b^2c^2} =abc$, which is the volume. So in this case $\sqrt{189L}$ is an integer, so $189L$ is a perfect square. Since $189=7 \cdot 3^3$, the smallest such possible value of L is $3 \cdot 7 =21$ $\blacksquare$