Wikipedia's article on Weyl groups shows an example of a root system and the corresponding fundamental chambers (in my understanding, also known as fundamental regions or fundamental domains). In this picture, the fundamental chambers contain some of the roots. I would expect the roots to determine the walls of the fundamental chambers and, therefore, not be contained in them.
Is this correct or am I missing something?
The picture is ok, I think that you might be misinterpreting or misremembering the way in which the roots determine the walls of the chambers. Recall the definition of the Weyl Chambers: A Weyl Chamber is a connected component of $E\setminus\bar{H}$, where $E$ is your underlying Euclidean space, and $\bar{H}$ is the union of the hyperplanes that are perpendicular to the roots (i.e., a hyperplane $H_\alpha$ in $\bar{H}$ is perpendicular to the line formed by joining $\alpha$ and $-\alpha$ for some root $\alpha$).
In the case of the root system $A_2$ (the Wikipedia picture), each hyperplane $H_\alpha$ will avoid all other roots. As Asal said, a root $\alpha$ will be in a wall iff there is a root $\beta$ such that $\langle\alpha,\beta\rangle = 0$ (i.e., imagining the roots as lines, they're perpendicular). This will not always happen, but it is possible. For example, in the root system $G_2$ some roots will indeed lie in some of the walls (in fact, in $G_2$ every root lies in a wall):