@CharlesRezk Hang on -- a 0-truncated object $X$ is one such that $Hom(Y,X)$ is discrete for all $Y$. For example $Hom(G,X)$ -- the underlying space of $X$ -- is discrete. Then since we are working in the Borel category, and $X \to \pi_0(X)$ is an underlying equivalence, it is an equivalence. So $X$ is a $G$-set, and therefore a disjoint union of orbits. No?

I was confused for a minute because $EG$ is an example of a Borel $G$-space which you might not want to think about as an orbit, but in fact it's the terminal object in Borel $G$-spaces, so it is equivalent to the orbit $G/G$.