How to calculate the correlation coefficient from minimal distributional assumptions?

Let random variables $X_1,X_2,\ldots,X_n$ satisfy $$(X_i,X_j)\stackrel{d}{=}(X_1,X_2)\quad \forall i, j$$

(that is, these variables are identically distributed and all their bivariate marginal distributions are the same, too) and suppose $$\sum_{i=1}^n X_i=0.$$

How can the correlation coefficient of $X_1, X_2$ be calculated?

The correlation coefficient $\rho$ will exist provided the $X_i$ have a finite nonzero variance (which easily implies $n \ge 2$ because in case $n=1,$ $X_1=0$ has zero variance). Since $X_1$ and $X_2$ have the same distribution their variances are equal, too, whence

$$\rho = \frac{\operatorname{Cov}(X_1,X_2)}{\sqrt{\operatorname{Var}(X_1)}\sqrt{\operatorname{Var}(X_2)}} = \frac{\operatorname{Cov}(X_1,X_2)}{\operatorname{Var}(X_1)}.\tag{*}$$

Take the one numerical fact you have--namely, the $X_i$ sum to zero--and compute a variance, using the assumption of identical joint distributions to replace all occurrences of $\operatorname{Var}(X_i)$ by $\operatorname{Var}(X_1)$ and $\operatorname{Cov}(X_i, X_j)$ by $\operatorname{Cov}(X_1, X_2):$

$$0 = \operatorname{Var}(0) = \operatorname{Var}(X_1 + \cdots + X_n) = n \operatorname{Var}(X_1) + n(n-1)\operatorname{Cov}(X_1,X_2).$$

Dividing both sides by $n\operatorname{Var}(X_1)$ and substituting $(*)$ yields the equation

$$0 = 1 + (n-1)\rho$$

whose solution is easily found.