Which roots of irreducible quartic polynomials are constructible by compass and straightedge?

What you have to check is that if $\alpha$ is constructible then all its conjugates are constructible. From here, using composite fields, you should prove that $\alpha$ lies in a finite Galois extension of degree a power of $2$.

Conversely, if a number $\alpha$ lies in a Galois extension of degree a power of $2$, it is constructible.

Therefore the constructible numbers are those for which the Galois group of their minimal polynomial is of order a power of $2$.

Since you know the possiblilities for the Galois group of an irreducible of degree $4$, you should have the answer.