I'm working on this pull request for the pi-Base and need to add peer reviewed counterexamples to the theorem that all $T_3$ Lindelöf spaces are realcompact, that is, a closed subset of some (not necessarily finite) power of the Euclidean line $\mathbb R$.
What such spaces exist?
The definition of $X$ being realcomapact iff $X$ can be embedded as closed subset of $\mathbb{R}^\kappa$ for some cardinal $\kappa$ is really unhandy to work with, so let me introduce another equivalent definition.
Definition. A $z$-ultrafilter $\mathcal{U}$ on $X$ is an ultrafilter on the lattice $Z(X)$ of zero sets of $X$. We say that it's real when it's closed under countable intersections (those ultrafilters would be called countably complete in set-theoretic language). We say that it's fixed when $\bigcap\mathcal{U}\neq\emptyset$, that is $\mathcal{U} = \{Z\in Z(X) : x\in Z\}$ for some $x\in X$ (from maximality). We say that $X$ is realcompact when every real $z$-ultrafilter on $X$ is fixed.
Here's a beautiful argument that discrete spaces of size continuum are realcompact.
Theorem. If $X$ is discrete with $|X| = \mathfrak{c}$ then $X$ is realcompact.
Proof: Consider $X = [0, 1]$ (with not necessarily the same topology). If $\mathcal{U}$ is a real ultrafilter on $X$, let $x\in [0, 1]$ be limit of $\mathcal{U}$ in Euclidean topology. If $U_n$ is a neighbourhood basis of $x$ in Euclidean topology, it follows that $U_n\in\mathcal{U}$ since $\mathcal{U}$ converges to $x$, and since $\mathcal{U}$ is real, its closed under countable intersections so that $\bigcap_n U_n = \{x\}\in \mathcal{U}$. This proves that $\mathcal{U}$ is a fixed ultrafilter, so $X$ is realcompact. $\square$
Another argument is based on the following that I won't prove:
Theorem. If $f:X\to Y$ is a continuous bijection between Tychonoff spaces and every subspace of $Y$ is realcompact, then every subspace of $X$ is realcompact.
In particular, since $\mathbb{R}$ is hereditarily Lindelof, and any Lindelof Tychonoff space is realcompact, any subspace of $\mathbb{R}$ is realcompact. So any subspace of discrete space $X$ of size $\mathfrak{c}$ is realcompact by taking $f:X\to\mathbb{R}$ to be any bijection, i.e. any discrete space of size $\leq \mathfrak{c}$ is realcompact.
In general we have
Theorem. Let $X$ be discrete, then $X$ is realcompact iff $|X|$ is non-measurable.
This follows pretty directly from definition of realcompactness (if you define it using $z$-ultrafilters anyway). Again we obtain that a discrete space of size $\mathfrak{c}$ needs to be realcompact. This is of course not Lindelof.
Other examples are $\mathbb{R}^\kappa$ for $\kappa\geq \aleph_1$, and any metrizable space of non-measurable cardinal that isn't second-countable (safe to assume all explicit examples of metric spaces are realcompact, since it's a set-theoretic issue whetever or not measurable cardinals exist).
EDIT: I'm leaving this up to illustrate why my naive argument below is incorrect: the set $D$ is not closed as $z\in\mathbb R^\kappa$ defined by $z(\alpha)=0$ always is a limit point of $D$: given any open neighborhood $U$ of $z$, only finitely-many coordinates $F\subseteq\kappa$ are restricted by $U$, so choose $\alpha\in\kappa\setminus F$ and note $f_\alpha\in U$ since $f_\alpha\upharpoonright(\kappa\setminus F)=z\upharpoonright(\kappa\setminus F)$.
Every discrete space is realcompact. In particular, let $\kappa$ be a cardinal and consider the subspace $D=\{f_\alpha:\alpha<\kappa\}\subseteq\mathbb R^\kappa$ where $f_\alpha(\beta)=\begin{cases}1\text{ if }\alpha=\beta\\0\text{ otherwise}\end{cases}$. This space is discrete: given $f_\alpha$, $\prod_{\beta<\kappa}U_\beta$ defined by $U_\beta=(0.5,1.5)$ when $\alpha=\beta$ and $U_\beta=\mathbb R$ otherwise is an open subset of $\mathbb R^{\kappa}$ only containing $f_\alpha$. This set is also closed. Thus $D$ is a discrete space of cardinality $\kappa$ which is a closed subspace of $\mathbb R^{\kappa}$.
Then when $\kappa\geq\aleph_1$ this space is not Lindelöf (the open cover of singletons has no countable subcover), giving an example for the desired question.