The answer is $\boxed{\frac{1}{6}}$.
To solve this question, we can use the rule of three. The rule of three states that if three quantities are in proportion, then the fourth quantity is also in proportion to the first three. In other words, if $a:b::c:d$, then $a:c=b:d$.
In this question, we are given that $\frac{1}{4}:\frac{1}{8}::\frac{1}{3}:x$. We can set up a proportion as follows:
$$\frac{1}{4}:\frac{1}{8}=\frac{1}{3}:x$$
Cross-multiplying, we get:
$$\frac{1}{4}x=\frac{1}{8}\times\frac{1}{3}$$
$$x=\frac{1}{8}\times\frac{1}{3}\times\frac{4}{1}$$
$$x=\frac{1}{6}$$
Therefore, the answer is $\boxed{\frac{1}{6}}$.
We can also solve this question by multiplying the two ratios together. The product of two ratios is the same as the ratio of the products of the corresponding terms. In other words, if $a:b::c:d$, then $a\times c:b\times d$.
In this question, we are given that $\frac{1}{4}:\frac{1}{8}::\frac{1}{3}:x$. We can multiply the two ratios together as follows:
$$\frac{1}{4}\times\frac{1}{3}:\frac{1}{8}\times x=\frac{1}{12}:x$$
$$x=\frac{1}{12}\times\frac{8}{1}=\frac{1}{6}$$
Therefore, the answer is $\boxed{\frac{1}{6}}$.