The ratio of the frequencies of two simple pendulums is $4: 3$ at the same place. The ratio of their respective lengths is

The frequency of a simple pendulum is given by $f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$, where $g$ is constant for both pendulums at the same location.

For two pendulums with frequencies $f_1$, $f_2$ and lengths $L_1$, $L_2$, the frequency ratio simplifies to:

$\frac{f_1}{f_2} = \sqrt{\frac{L_2}{L_1}}$

Given $f_1 : f_2 = 4 : 3$, we have $\frac{4}{3} = \sqrt{\frac{L_2}{L_1}}$.
Squaring both sides yields $\left(\frac{4}{3}\right)^2 = \frac{L_2}{L_1}$, so $\frac{16}{9} = \frac{L_2}{L_1}$.

The length ratio is therefore $L_1 : L_2 = 9 : 16$.

Final answer: $\boxed{\text{C}}$