Such fields are related to real-closed fields, which satisfy even a stronger property. A field $K$ is called real-closed, if it is formally real, i.e., $-1$ is a sum of squares in $K$, and no proper algebraic extension is formally real. Then we have the following result:

Theorem: In a real-closed field, every polynomial of odd degree $>1$ has a root.

As a consequence, in a real-closed field, every polynomial splits into linear and quadratic factors. Of course, "cubically closed fields" are more general than real-closed fields, but nevertheless this might give a direction to search for characterisations, and there is a large literature on real-closed fields.

Such fields are related to real-closed fields, which satisfy an even stronger property. A field $K$ is called real-closed, if it is formally real, i.e., $-1$ is not a sum of squares in $K$, and no proper algebraic extension is formally real. Then we have the following result:

Theorem: In a real-closed field, every polynomial of odd degree $>1$ has a root.

As a consequence, in a real-closed field, every polynomial splits into linear and quadratic factors. Of course, "cubically closed fields" are more general than real-closed fields, but nevertheless this might give a direction to search for characterisations, and there is a large literature on real-closed fields.