The modulus of the square root of the conjugate of $-7+24 \sqrt{-1}$ is ....

The complex number is $z = -7 + 24i$, so its conjugate is $\bar{z} = -7 - 24i$.

The modulus of $\bar{z}$ is calculated as $|\bar{z}| = \sqrt{(-7)^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25$.

Using the property $|\sqrt{w}| = \sqrt{|w|}$ for any complex $w$, we find $|\sqrt{\bar{z}}| = \sqrt{|\bar{z}|} = \sqrt{25} = 5$.

Alternatively, solving $\sqrt{-7 - 24i} = x + iy$ leads to $(z = 3 - 4i)$ or $(z = -3 + 4i)$, both of which have modulus $5$.

Final answer: $\boxed{5}$