Considering $M$ as a linear operator from $\mathbb R^n$ to $\mathbb R^m$, its range $\text{Ran}(M)$ is a linear subspace. Linear subspaces in finite-dimensional spaces are closed. Thus if $b \notin \text{Ran}(M)$, its distance to $\text{Ran}(M)$ is nonzero, i.e. there is $\epsilon > 0$ such that $\|M x' - b\| > \epsilon$ for all $x' \in \mathbb R^n$.

Considering $M$ as a linear operator from $\mathbb R^n$ to $\mathbb R^m$, its range $\text{Ran}(M)$ is a linear subspace. Linear subspaces in finite-dimensional spaces are closed. Thus if $b \notin \text{Ran}(M)$, its distance to $\text{Ran}(M)$ is nonzero, i.e. there is $\epsilon > 0$ such that $\|M x' - b\| > \epsilon$ for all $x' \in \mathbb R^n$.

We can be a little bit more quantitative, perhaps. We can use $M$ to construct an orthogonal projection $P$ on $\text{Ker}(M^{T}) = (\text{Ran}(M))^\perp$. If $b \notin \text{Ran}(M)$, then $0 < \|Pb\| \le \|b - M x'\|$ for all $x' \in \mathbb R^n$.