가족이 뭐 별건가? = 가족이 뭐 별것인가? = Is family such a special thing? 별것 (별 = different; particular; special, 것 = thing) means something special or peculiar. E.g. A: 손에 들고 있는 거 뭐야? What's that you've got in your hand? B: 응, 별거 아니야. It's nothing / It's nothing interesting. "사랑이 벌거더냐? 좋아하면 사랑이지." = Is love some fancy thing? No, it just means you like someone. (from an old song 짚세기 신고 왔네, "I Came in Straw Shoes")

별건가 means "It is nothing special to mention."