I recently came upon this question in the context of distributions taking values in a finite set, but since yours take values in the compact interval $[0,1]$ I don't think much will go wrong applying the answer to your setting.

Certainly a sufficient condition is that you can construct an exchangeable sequence $\chi_1,\chi_2,\ldots$ for which the marginal of $\chi_1$ and $\chi_2$ is $\Xi$, or equivalently that $\Xi$ is a mixture of i.i.d. distributions per de Finetti's theorem. It turns out this is also necessary.

To see this suppose that $\Xi$ satisfies the condition you give on its two-variable marginals. Then for any finite $n$ there is a finite sequence of random variables $\xi_1,\ldots,\xi_n$ (just take the first $n$ variables of your given sequence) all of whose two-variable marginals are $\Xi$. We can construct a new sequence of random variables $\chi_1,\ldots,\chi_n$ by randomly permuting the $\xi_1,\ldots,\xi_n$. By linearity the marginal of any two of these will still be $\Xi$.

But the distribution of the $\chi_1,\ldots,\chi_n$ is invariant under arbitrary permutations, by symmetry. Using a diagonalization and compactness argument, we get from the existence of such a sequence $\chi_1,\ldots,\chi_n$ for all finite $n$ the existence of an exchangeable sequence $\chi_1,\chi_2,\ldots$ whose two-variable marginal distribution is $\Xi$.