뜨다 here means "to open". I don't agree with Keon-seop, 눈코 does refer to the eyes and the nose (it's one word so no space). 눈코 뜰 새 없이 바쁘다 = to be so busy that you don't have the time to open your eyes and nose Of course you don't really open your nose, but it's just a fixed expression. :)

눈 = eyes? , no it's not 코 = nose ? , no 뜰(뜨다 +ㄹ)? no,no! In this sentense 눈 and 코 means parts of a net. 뜨다 means to sew . then, 눈 코 뜰 새 없이 바쁘다. means 'too busy to repair the net'