Correlation coefficient can be interpreted as the cosine of the angle, $\theta$ between the centered random variables in a vector space. When $\cos \theta = \rho = 1$, we have $\theta = 0$. When $\cos \theta = \rho = -1$, we have $\theta = \pi$.
Because all $n$ random variables have the same pairwise correlation, I believe the range of $\rho$ is limited to $1$ only. The angle between the random variables in vector space can only be $0$, otherwise we would contract the given constraint that all pairwise correlation coefficients are the same.
Is this the right idea?