It needn't have everywhere good reduction, no. For instance if $C_{/\mathbb{Q}}$ is a curve which has semistable bad reduction at a prime $p$, then this reduction stays semistable bad over any algebraic extension.
For easy examples, take any modular curve $X_0(p)$ for any prime number $p \geq 23$. Such curves have genus at least $2$ and semistable bad reduction at $p$.
More generally, it is easy to construct for each $g \geq 2$ and prime number $p$ a curve $C_{/\mathbb{Q}}$ of genus $g$ with semistable bad reduction at $p$. Examples of curves over $\mathbb{Q}_p$ (and also defined over some number field $K$ and thus also over $\overline{\mathbb{Q}}$) with these properties (and others...) are constructed in this paper of mine.