Typescript return type for Observables in getter and setter

I think you're almost there but I question using the setter, at least in the code you have provided. When you look at the code inside your setter you're actually not setting anything to _user$.

I think this is what is tripping you up, I don't think you should be using a setter here. Based on what the code is doing inside your setter, this should just be a public function. This is what I think it should look like

constructor(private http: Http) {
  this.storage = new Storage(SqlStorage);
  this._user$ = new Subject();
  this.dataStore = { user: {} };
}

setUser(user: User) {
  this.storage.set('user', JSON.stringify(user));
  this.dataStore.user = user;
  this._user$.next(this.dataStore.user);
}

get user$() {
  return this._user$.asObservable();
}

Then your component that is using the Service, would call UserService.setUser(userObj).

When you follow errors messages, you will have only two pairs of setter and getter:

get var(): User {}
set var(user: User) {}

And

get var(): Observable<User> {}
set var(user$: Observable<User>) {}

Actually that are the only allowed combinations.