DUE DILIGENCE: I have reviewed the list of questions possibly related to the one that I pose below, and I find that none of them address my particular question.
Is there a formal definition for the class of non-cubic integers whose cube roots are constructible using a compass and a MARKED straightedge?
If such a class exists, then $2$ would be a member of it. Are there other known members not derivable from $2$?
In fact, cube root of any number $k$ can be constructed given an interval of length $k$, marked ruler and compass. See here for such a construction along with a proof.