Angular Momentum Of S-subshell of an atom?
Your confusion arises from the wrong choice of equation for the angular momentum. Your equation

\begin{align}
\lVert \vec{l} \rVert = m \lVert \vec{v} \rVert \lVert \vec{r} \rVert \ ,
\end{align}

where $\vec{l}$ denotes the angular momentum vector ($\lVert \vec{l} \rVert$ is its length, i.e. the angular momentum), $\vec{v}$ the velocity vector, $\vec{r}$ the radius vector  and $m$ the mass, does already entail some assumptions that are not necessarily valid. It derives from the more general equation

\begin{align}
\vec{l} = \vec{r} \times  \vec{p} ,
\end{align}

where $\vec{p}$ is the momentum vector which under certain circumstances can indeed be written as $\vec{p} = m \vec{v}$ (there are a few subtleties that I don't want to touch here, keyword: canonical momentum, since they don't relate to the problem at hand) and $\times$ denotes the cross product between two vectors. If you now take the absolute value of the angular momentum vector and use the properties of the cross product you arrive at

\begin{align}
\lVert \vec{l} \rVert &amp;= \lVert \vec{r} \times  \vec{p} \rVert \\
&amp;= \lVert \vec{r} \rVert \lVert \vec{p} \rVert \sin \theta \ ,
\end{align}

where $\theta$ is the angle between the vectors $\vec{r}$ and $\vec{p}$. So, the factor that is missing in your equation is the $\sin \theta$ term, whose absence means that your equation implicitly assumes that $\theta = 90°$ since $\sin 90° = 1$. And indeed if this would be the case, then for $ \lVert \vec{l} \rVert = 0$ to be true either $ \lVert \vec{r} \rVert$ or $ \lVert \vec{p} \rVert$ had to be equal to zero. However, since this assumption isn't made anywhere in the quantum mechanical description of an atom the $\sin \theta$ term has to be kept and there is another way to achieve $\lVert \vec{l} \rVert = 0$, namely $\vec{r}$ and $\vec{p}$ being either parallel ($\theta = 0°$) or antiparallel ($\theta = 180°$) to each other because then the $\sin \theta$ term is equal to zero.

Of course, for the quantum mechanical description you have to replace the vectors by the quantum mechanical operators and the absolute values of the vectors by the expectation values of the operators.