Él se hizo cargo de los negocios de la familia después de la muerte de su padre.

Does this mean business like “the family affairs” or does this mean that the family owned actual businesses and the person took them over?

A good question! To shed a little light on the subject, here’s what DeepL Translate comes up with:

  • He took over the family business. “Se hizo cargo del negocio familiar.”
  • He took over the family affairs. “Se hizo cargo de los asuntos familiares.”
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English Translation

He took charge of the family business after his father died.

I think “hacerse cargo de” generally means “to take care of” or “to take charge of”