The value of $\sqrt{0.\bar{4}}$ is

The repeating decimal $0.\bar{4}$ can be expressed as a common fraction:
$$0.\bar{4}=\frac{4}{9}$$

Now, taking the square root of this value:

$$\sqrt{0.\bar{4}}=\sqrt{\frac{4}{9}}=\frac{2}{3}$$

Converting the fraction $\frac{2}{3}$ back to a repeating decimal:

$$\frac{2}{3}=0.6666\cdots =0.\bar{6}$$