The answer is $\boxed{\frac{1}{2}}$.
In the proportion $\frac{1}{4}:\frac{1}{8}::\frac{2}{3}:x$, the first and third terms are in the ratio $2:1$, and the second and fourth terms are in the ratio $4:1$. This means that the fourth term is equal to the product of the first and third terms, divided by the product of the second and third terms, or $\frac{1}{4}\times\frac{2}{3}=\frac{1}{2}$.
Option A, $\frac{1}{4}$, is incorrect because it is the first term of the proportion, not the fourth term. Option B, $\frac{1}{3}$, is incorrect because it is the second term of the proportion, not the fourth term. Option C, $\frac{1}{2}$, is correct because it is the product of the first and third terms, divided by the product of the second and third terms. Option D, $\frac{2}{7}$, is incorrect because it is not in the ratio $2:1$ with the first term of the proportion.