If you play a game with aging on and story progression off, townies will die of old age but will not be repopulated. After a few generations, your family will be the only one left, unless you go around and make all the other families have children yourself.
If it seems to only affect old games, here's a thought: Save the family in question to the bin, then evict them from the house and save the house to the bin separately. Start a new neighborhood, placing the house where it was previously and move the family from the bin into it. They will lose their previous family tree and all of their friends, but as I remember they retain their career levels.