Answer
See the detailed answer below.
Work Step by Step
We know that the penetration distance is given by
$$\eta = \frac{\hbar}{\sqrt{2m(U_0 - E)}}$$
Now we need to write the unit of each variable,
$$\eta \Rightarrow \rm \dfrac{J\cdot s}{\sqrt{kg\cdot J}}= \dfrac{\dfrac{kg\cdot m^2}{s^2}\cdot s}{\sqrt{kg\cdot \dfrac{kg\cdot m^2}{s^2}}}$$
$$\eta \Rightarrow\rm \dfrac{\dfrac{kg\cdot m^2}{s } }{ \dfrac{kg\cdot m }{s }}= \dfrac{kg\cdot m^2}{s } \cdot \dfrac{s }{kg\cdot m }=m$$
So the unit of the penetration distance is the meter.
