$\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\scrT{\mathscr T}\def\ssp{\kern.4mm}
$More specifically, **I ask** whether $S$ *be a Borel set in the topological space* $(\Omega,\scrT)$ in the following situation.

Let $\Omega$ be the set of all real analytic functions $x:\bbR\to\bbR$, and let $X$ be the abstract real vector space obtained when $\Omega$ is equipped with the "obvious" pointwise operations. Let $S$ be the set of all $x\in\Omega$ having a holomorphic extension $\bbC\to\bbC$. For any open set $U$ in $\bbC$ with $\bbR\subseteq U$, let ${\rm F}\,U$ be the real Fréchet space of analytic functions $x:\bbR\to\bbR$ possessing a holomorphic extension $U\to\bbC$, equipped with the topology induced from the compact-open topology of the space of holomorphic functions $U\to\bbC$. Now let $\scrT$ be the strongest locally convex topology for $X$ such that for $F=(X,\scrT)$ and for all the above sets $U$ the identity is a continuous linear map ${\rm F}\,U\to F$.

Letting ${\rm F}_\bbC\,U$ denote the similar real spaces of complex valued real analytic functions, and using Runge's theorem in conjunction with the open mapping theorem applied to the linear map ${\rm F}\,U\times{\rm F}\,U\to{\rm F}_\bbC\,U$ given by $(x,y)\mapsto x + {\rm i}\,y$, if I am not mistaken, one can show that $S$ is a dense linear subspace in $F$.

**Added.** By Robert Israel's answer we have $S=\bigcap\ssp\big\{\,\bigcup\ssp\big\{\,\bigcap\kern1mm\{\,\{\,x\in\Omega:|\,a_n(x)\,|^{\,n^{\ssp-1}} < j^{\ssp-1}\,\} : k < n \in\mathbb Z\kern2mm\} : k\in\mathbb N_0 \,\} : j\in\mathbb N\kern2mm\} \kern1mm$, a Borel set since every $x\mapsto a_n(x)=(\ssp n\ssp!\ssp)^{\ssp-1}\,{\rm D}^{\,n\ssp}x\ssp(0)$ is a continuous linear functional on $F\,$.