One can see a counterexample easily for the reals $\mathbb{R}$ if the Continuum Hypothesis holds, for in this case the reals $\mathbb{R}$ are the union of a chain of countable sets. Simply well-order the reals in order type $\omega_1$ and for countable ordinals $\alpha$ let $X_\alpha$ be the first $\alpha$ many points in this enumeration. So every $X_\alpha$ is countable, but the union is all of $\mathbb{R}$.

More generally, avoiding the CH assumption, let $\kappa$ be the cardinality of the smallest non-measure $0$ set $X$ of reals. We can enumerate $X$ in order type $\kappa$, and the initial segments of this enumeration all have measure $0$, but the union is $X$, which is non-measure $0$.

The cardinal $\kappa$ used above is known as the *uniformity number* for Lebesgue measure in the theory of cardinal characteristics; for example, see this MO answer or Andreas Blass' handbook article.