Let us work with separable metrizable spaces.
It's known that if $X$ contains a dense completely metrizable subspace then $X$ is Baire.
Moreover, if we assume that $X$ embeds into any space as a Borel set ($X$ is absolutely Borel), then it contains a dense completely metrizable subspace.
What's an example of a Baire space which doesn't contain a dense completely metrizable subspace?
Take a Bernstein set $B$ in the reals, as $B \subseteq \Bbb R$, it's separable metric. The defining property for $B$ is that it intersects $P$ for any uncountable closed subset of the reals, but also $\Bbb R\setminus P$.
There are no constructive examples of such a set, and they are proved to exist by well-ordering the reals and doing a transfinite recursion. If you believe in AC, they do exist.
Then $B$ is a Baire space (not too hard to show, e.g. use Cantor's intersection theorem) but it doesn't contain a completely metrisable subspace (as these always contain Cantor sets, and these cannot be a subset of $B$).
A nasty beast, but they do exist.