Some SDE's even though they are defined on $\mathbb R$ have solutions that are confined to a region. One example with explicit solution could be:
$$ dX_t = -\frac{1}{2} X_t (1-X_t^2) dt + (1-X_t^2) dW_t $$
with $W_t$ standard Wiener process, and solution: $$ X_t = \tanh(W_t + \operatorname{arctanh} X_0) $$
This equation has impermeable barriers at $X=1$ and $X=-1$ where both drift and noise terms are zero.
Is it enough for drift and noise terms to be zero for such barrier to exist?
Even though solutions of SDE's can intersect for different paths of the Wiener process, for one specific path of the Wiener process solutions are unique given initial condition (and thus cannot intersect). With that in mind $X_t = q = const$ for value of $q$ such that it is a root of both drift and noise term is a solution.
Now pick a path of the Wiener process and suppose we have solution that is nonconstant and intersects $X_t = q$ for some $t_0$. Then the remainder of such trajectory ($t>t_0$) would be an alternative (to the constant) solution starting at the same initial condition violating uniqueness of strong solutions.
To summarise: yes, zeros of both drift and noise term guarantee an impermeable barrier for the solutions of the SDE.
On a related note you can see that from another angle by changing SDE into a PDE for the evolution of the $X_t$ distribution w.r.t. $t$ (when possible) and see that at the barrier point probability cannot raise or fall ($\partial_tp(x,t)$), and is initially zero so it shouldn't spill over through such barrier.