You need two tables, A Family table where an auto-entered serial number produces your unique family code and a Person table where you have one record for each person.
You can link the two tables by Family Code:
Family::FamilyCode = Persons::FamilyCode
You can place a portal to Persons on your Family layout and use it to list all members of a given family and if you enable "Allow creation of records via this relationship" for the Persons table in this relationship, you can add family members simply by typing in the portal.
If, thanks to modern family issues, you need to list someone as a member of more than one family, you may need a third table to serve as a "Join" table to enable you to make such links without entering the same person's information twice.