Hibernates stores all properties that declare a type that implements Serializable.
But the type of your days property is a Set, which does not extend Serializable. To store your set as serialized value you will need to declare a set class which implements Serializable - like most do. For example HashSet:
@Entity
public class SomeEntity {
private HashSet<DayOfWeek> days;
}
You don't need any further annotations. This works for Enum and all other serializable elements.
But for a set of enums exists another option: Use an EnumSet, which stores the values into an integer column (every enum value is exactly one bit). See my answer here: JPA map collection of Enums