The original problem solves in the positive: there is a model of ZFC in which there exists a countable OD (well, even lightface $\Pi^1_2$, which is the best possible) set of reals $X$ containing no OD elements. The model (by the way, as conjectured by Ali Enayat at http://cs.nyu.edu/pipermail/fom/2010-July/014944.html) is a $\mathbf P^{<\omega}$-generic extension of $L$, where $\mathbf P$ is Jensen's minimal $\Pi^1_2$ real singleton forcing and $\mathbf P^{<\omega}$ is the finite-support product of $\omega$ copies of $\mathbf P$. 

**A few details.** Jensen's forcing is defined in $L$ so that $\mathbf P =\bigcup_{\xi<\omega_1} \mathbf P_\xi$, where each $\mathbf P_\xi$ is a ctble set of perfect trees in $2^{<\omega}$, *generic* over the outcome $\mathbf P_{<\xi}=\bigcup_{\eta<\xi}\mathbf P_\eta$ of all earlier steps in such a way that any $\mathbf P_{<\xi}$-name $c$ for a real ($c$ belongs to a minimal countable transitive model of a fragment of ZFC, containing $\mathbf P_{<\xi}$), which $\mathbf P_{<\xi}$ forces to be different from the generic real itself, is pushed by $\mathbf P_{\xi}$ (the next layer) not to belong to any $[T]$ where $T$ is a tree in $\mathbf P_{\xi}$. The effect is that the generic real itself is the only $\mathbf P$-generic real in the extension, while examination of the complexity shows that it is a $\Pi^1_2$ singleton. 

Now let $\mathbf P^{<\omega}$ be the finite-support product of $\omega$ copies of $\mathbf P$. It adds a ctble sequence of $\mathbf P$-generic reals $x_n$. A version of the argument above shows that still the reals $x_n$ are the only $\mathbf P$-generic reals in the extension and the set $\{x_n:n<\omega\}$ is $\Pi^1_2$. Finally the routine technique of finite-support-product extensions ensures that $x_n$ are not OD in the extension.

**Addendum.** For detailed proofs of the above claims, see [this manuscript][1].

Jindra Zapletal informed me that he got a model where a $E_0$-equivalence class $X=[x]_{E_0}$ of a certain Silver generic real is OD and contains no OD elements, but in that model $X$ does not seem to be analytically definable, let alone $\Pi^1_2$. The model involves a combination of several forcing notions and some modern ideas in descriptive set theory recently presented in *Canonical Ramsey Theory on Polish Spaces*. Thus whether a $E_0$ class of a non-OD real can be $\Pi^1_2$ is probably still open.


  [1]: http://xxx.tau.ac.il/pdf/1408.3901.pdf