Can you set a static enum inside of a TypeScript class?

To do this, you would need to define it outside of the class first, then assign it as a static property.

enum UNIT_STATUS {
    NOT_STARTED,
    STARTED,
    COMPLETED,
}

class UnitModel extends Backbone.Model {

    static UNIT_STATUS = UNIT_STATUS;

    isComplete(){
        return this.get("status") === UNIT_STATUS.COMPLETED;
    }
}

export = UnitModel;

You can declare a namespace right after your class, and declare the enum inside the namespace. For example:

class UnitModel extends Backbone.Model {
    defaults(): UnitInterface {
        return {
            status: UNIT_STATUS.NOT_STARTED
        };
    }


    isComplete(){
        return this.get("status") === UNIT_STATUS.COMPLETED;
    }

    complete(){
        this.set("status", UNIT_STATUS.COMPLETED);
    }
}

namespace UnitModel {
    export enum UNIT_STATUS {
        NOT_STARTED,
        STARTED,
        COMPLETED
    }
}

export = UnitModel;

Then you can use UnitModel.UNIT_STSTUS to reference you enum.