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")