  We prove that rationally connected Calabi--Yau 3-folds with kawamata log
terminal (klt) singularities form a birationally bounded family, or more
generally, rationally connected $3$-folds of $\epsilon$-CY type form a
birationally bounded family for $\epsilon>0$. Moreover, we show that the set of
$\epsilon$-lc log Calabi--Yau pairs $(X, B)$ with coefficients of $B$ bounded
away from zero is log bounded modulo flops. As a consequence, we deduce that
rationally connected klt Calabi--Yau $3$-folds with mld bounded away from $1$
are bounded modulo flops.
