"My family is pretty good recently."
很好 does not necessarily mean very good, as 很 is generally used and therefore isn't particularly strong as an adverb. So I see no reason why saying 很好 as opposed to 不错 is an issue. Also, would 家 not suffice as a translation to family? Perhaps it isn't as common/clear as 家人, but the latter refers specifically more to the family members as opposed to the family as a whole, no?