This is a mare coincidence. The Slavic word for family comes from the same root that gave the English home. Back in the pre-historic times it was pronounced with a palatal ḱ in the beginning, which in the Germanic, Italic, Celtic, and Greek languages equalized with the ordinary k, while in the Balto-Slavic and in the Indo-Iranian it has evolved to a sibilant sound - in Russian, in particular, it has become just s.
PS семья is unrelated with семя (seed), either. The later comes from the word for "sowing".
Note: Since “big” is here used as a predicate adjective, one would ordinarily use the predicate (short) form in Russian. But «большой|большое|большая|большие» is exceptional in not having its own predicate form «*бóлеш|*бóльшo|*большá|*больши́». So instead, one either uses the attributive (long) form, as in the given translation, or substitutes that of «великий|великое|великая|великие»=“great”: «вели́к|велико́|велика́|велики́».