There's an equivalent characterization for stein manifold:
A complex manifold is $\textbf{Stein}$ if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.( refer to Hormander's standard textbook e.g.)
However,as well known,the unit ball in $\mathbb{C}^m$ is a stein manifold .So,The two facts are contradictory,the unit ball in $\mathbb{C}^m$ cannot be a closed submanifold of some $\mathbb{C}^N$.
I want to know, what's wrong?
The two facts are not contradictory. Unit ball in $ \mathbb{C}^m $ can be properly embedded into $\mathbb{C}^{2m+1}$ by the theorem 5.3.9 in Hormander's book even though it is open in $ \mathbb{C}^m $.