I am learning quantum field theory. I understand that the solution of Dirac equation has four states and each corresponds to a spinor. These four states are exactly the eigenstates of the spin operator and their eigenvalues are +1/2 or -1/2. But it seems that I can construct another operator (matrix), may be 8 by 8 or some other dimensions. The corresponding eigenvalues of this matrix can of course be 1/4, 1/2, 3/4, etc. With these eigenstates I can also construct a formula and make the eigenstates the solution of this formula. Then I can have a spin 1/4 spinor. I don't understand why Dirac equation can be so special.
There is no such thing as spin $1/4$ in 4-dimensional spacetime.
Spin has to do with representations of the Lorentz Lie algebra. So let me shed some light on how these are classified.
First, the complex-valued Lorentz algebra $so(1,3)$ is equivalent to $so(4)$. The $so(4)$ algebra is equal to the direct sum of two copies of $su(2)$, which means that irreducible representations of $so(1,3)$ are labeled by ordered pairs of the irreducibles of $su(2)$.
The $su(2)$ representation theory can be found in any textbook on Lie groups or even in some QFT textbooks. One of the crucial facts is that irreducibles of $su(2)$ are labeled by nonnegative half-integers called spins:
$$ j = 0, \; 1/2, \; 1, \; 3/2, \; 2, \; 5/2, \; \dots $$
This is enough to start constructing irreducibles of the Lorentz algebra. The basic blocks of the representation theory are the two fundamental representations $(1/2, 0)$ and $(0, 1/2)$ which are called the left and right Weyl spinors respectively. Both are two-dimensional.
Dirac spinors actually belong to the 4-dimensional reducible representation $$ (1/2, 0) \oplus (0, 1/2). $$
Another 4-dimensional representation is the irreducible $(1/2, 1/2)$ to which the 4-vectors belong.
As you can see, this is all just representation theory and there is no spin-$1/4$ representation in 4 spacetime dimensions.
However, in $2$ spacetime dimensions this is no longer valid and representations with fractional spins exist.