Question from Introduction to Modern Cosmology by Andrew Liddle
From the definition of the rms (e.g. here),
$$
\mathrm{rms}^2(x) = \langle x \rangle^2 + \sigma_x^2,
$$
where $\langle x \rangle$ is the mean value, and $\sigma_x$ is the dispersion.

For galaxies with random velocities, the mean velocity should be $\langle x \rangle = 0$, unless they drift in some direction. Hence, the velocity dispersion should be $600\,\mathrm{km}\,\mathrm{s}^{-1}$.

For randomly distributed velocities, the speed $V_{||}$ you measure along your line of sight (LOS) is a factor $\sqrt{3}$ smaller, i.e. $\sim350\,\mathrm{km}\,\mathrm{s}^{-1}$.

Thus, to get a 10% accuracy, the mean value should be 10 times larger, i.e. $\sim 3500\,\mathrm{km}\,\mathrm{s}^{-1}$.

For a Hubble constant of $H_0 = 100\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, that corresponds to a distance of
$$
d = \frac{v_{||}}{H_0} \simeq 35\,\mathrm{Mpc},
$$
while for $50 \,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ you get $70\,\mathrm{Mpc}$.