"우리사이에 거리만 멀뿐이야." => 우리 사이에 거리만 멀 뿐이야. (after word spacing correction) => 우리 사이의 거리만 멀어질 뿐이야. (more clear) => (literally) Only the distance between us will become greater. => The only thing to happen is for us to grow farther apart. 멀 뿐 is 멀다(far) + 뿐(이다) (only), as chin.billy.leung explained. But in this sentence 멀어지다 (become more distant) should make the meaning clearer, as it's talking about how distant the two will become. This is another of those poorly written, throw-away sentences that abound on the Internet nowadays. For learning purposes, it is always better to concentrate on good materials.

This is not one word at all. It comes from 멀다 + ㄹ 뿐. The ㄹ actually gets absorbed into 멀다. There should be a spacing in between 멀 and 뿐 to be strictly pedantic. If you look up the dictionary you should see the latter means (only, simply). The former means far apart which obviously is referring to the relation part in the beginning.