Consider a singleton space $\{x\}$, it is a manifold and it is locally euclidean as there is a homeomorphism to $\mathbb{R}^0$. However, consider $S^0=\{-1,1\}$ with the discrete topology, there does not exist any homeomorphism to any open neighbourhood of $\mathbb{R}^k$ for $k \ge 0$, so it cannot be locally euclidean and cannot be therefore a manifold.
So is $S^0$ the most trivial example of a topological space that is not a manifold?
'Locally Euclidean' means each point has a neighborhood that is homeomorphic to $\mathbb R^k$. Each point in $S^0$ does have this property for $k = 0$.
More generally, any countable discrete space is a $0$-manifold.