Translation:My youngest sister is a teacher, my oldest a doctor.
It sounds ok without "one" to my british ears, and correct but a bit foreign with "one". But i would normally say eldest, rather than oldest. Old seems to go "old, older, oldest" most times, and always good, but when talking of family members it seems quite ok to use "old, elder, eldest". If a parent talks of his oldest child it seems normal to omit reference to son, daughter, child, if you use "eldest". "My eldest is a doctor" to me means "my oldest child is a doctor". I would use eldest only in family relationships, not so much for unrelated people, eg classmates. Learning another language forces you to look at how you use your own, and my usage might be atypical, but i doubt it.