How is it possible to obtain a value of 8/3 as an oxidation state?
The formula of $\ce{Fe3O4}$ is a shorthand for $\ce{Fe^{II}Fe^{III}2O4}$, so there are two sorts of atoms of iron here in +2 and +3 oxidation state.

However, it is still possible to gain truly fractional oxidation states. The simplest example probably would be potassium superoxide, $\ce{KO2}$. Here, the anion is symmetrical $\ce{O2^{-}}$ anion, so obviously both oxygen atoms are in $\ce{-1/2}$ oxidation state. There are more complicated examples as well. 

Often fractional oxidation state comes from symmetrical structures with charge, that would have unpaired electrons otherwise. However, systems with truly fractional oxidation states usually have bonding, hard to describe in terms of common Lewis structures (and with little relevance of idea of oxidation state), and mostly are considered in semi-advanced organic chemistry and advanced inorganic chemistry.