This is rather a comment to Nate Eldredge's answer, which reduces the (negative) answer to the existence of a set $\Omega\subset \mathbb{R}^2$ which is not a countable intersection of countable unions of products: $$\Omega\ne \cap_{n=1}^\infty \cup_{k=1}^\infty B_{n,k}\times C_{n,k}.$$

Call a set $A\subset \mathbb{R}^2$ *kind-of-open*, if it is a countable union of product sets: $A=\cup_{n=1}^\infty B_n\times C_n,\quad B_n,C_n\subset \mathbb{R}$ (open sets are kind-of-open). A finite intersection and a countable union of kind-of-open sets is kind-of-open. Maybe there is some standard name for such sets?

Define a *finite rank function* $f\colon \mathbb{R}^2\to \mathbb{R}$ as a finite sum $f(x,y)=\sum_{n=1}^N g_n(x)h_n(y)$.

Note that if $f$ is a finite rank function and $U\subset \mathbb{R}$ is open, the preimage $f^{-1}(U)$ is kind-of-open. Indeed,
$$
f^{-1}(U)=\cup (\cap_{n=1}^N g_n^{-1}(\Delta_n))\times (\cap_{n=1}^N h_n^{-1}(\delta_n))
$$
where the union is taken over all tuples of rational intervals $\Delta_1,\dots,\Delta_n,\delta_1,\dots,\delta_n$ satisfying the inclusion
$\sum_{n=1}^N \Delta_n\cdot \delta_n\subset U$.

Note that an infinite sum $f=\sum_{n=1}^\infty g_n(x)h_n(y)$ is a pointwise limit of finite rank functions $f_n$ (namely, partial sums).

Now let $\Omega\subset \mathbb{R}^2$ be any set. Assume that the characteristic function $f$ of $\Omega$ is a pointwise limit of finite rank functions $f_n$. Then $W_n=f_n^{-1}(1/2,\infty)$ is a kind-of-open set and so is $V_n=\cup_{k\geqslant n} W_k$, and $V_1\supset V_2\supset V_3\dots$. We have $\Omega=\cap_n V_n$. So it remains to find a set which is not a countable intersection of kind-of-open sets (kind-of-$G_\delta$ set).