The sharp general result in this direction is the classical [law of the iterated logarithm](https://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm) (LIL).  Suppose, after renormalizing if necessary, that the $x_n$ are iid with zero mean and unit variance.  Then the LIL states that
$$\limsup_{n \to \infty} \frac{X_N}{\sqrt{N \log \log N}} = \sqrt{2}, \quad \text{a.s.}$$
In your language, that says that with probability 1, $X_N$ is $O(\sqrt{N \log \log N})$, and that this cannot be improved to $O(\sqrt{N})$.  

To be more careful, it says that for $P$-almost every $\omega$, there is a finite number $C(\omega)$ such that $|X_N(\omega)| \le C(\omega) \sqrt{N \log \log N}$ for all $N$.