$\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 $F\,$.



Is the set of entire functions Borel in the space of analytic functions?
$f$ has an entire extension iff the radius of convergence of the Maclaurin series of $f$ is $+\infty$, i.e. iff $\limsup\_{n \to \infty} |a\_n|^{1/n} = 0$ where $a\_n$ are the coefficients of that series. The coefficients can 
be defined as limits of expressions defined in terms of point evaluations 
(finite-difference formulas), and those expressions are continuous functions on $\Omega$.
So yes, it is a Borel set, in fact an $F\_{\sigma\delta\sigma\delta}$.


