There should be a natural pause after the time indicator that begins the sentence (In the old days), so a comma is needed there. Then you have to combine an independent clause, family meant the people that you were related to, with a dependent clause, people who shared a bloodline and a common history. This requires some sort of punctuation for separation purposes (clarity). Semicolons are only used when combining two independent clauses, which these are not. The correct choice uses a colon because the second of these clauses gives more information about the first one.