Most people that have written about the meaning behind this song have taken it literally; because it must be a guy with cancer talking to his lover, right? I actually disagree.
I believe "Cancer" is actually metaphorical; symbolism, in a way, as in cancer is a representation of an illness he has. The vocalist is singing about something that is killing him; for sake of this point I'm going to say it was depression. This man is fighting his demons, and as the song depicts, is giving up on himself. He starts by saying "Turn away". Like most patients, people with mental illnesses often are ashamed of themselves. He's saying 'turn away' as in please, don't look at me, I'm broken. He continues; after talking about his death and what he wishes to happen, he sings "My sisters and my brothers, still, I will not kiss you. 'Coz the hardest part of this is leaving you." This is where the song takes a darker turn. He is officially giving up on himself (on the verge of suicide) as things have gotten too hard, and he doesn't want to 'kiss' them because he is not only ashamed but is done, and "the hardest part of this is leaving" them, his friends and family. "Now turn away, coz I'm awful just to see." Again, he is saying don't look at me, I'm broken, awful even to see. "Oh, my agony." He is tormented by his demons and is in agony, and combined with the previous lines he is almost saying don't look at me, I'm in pain, don't look at my agony. "Know that I will never marry." He's dying young, and this represents that. "Baby, I'm just soggy from the chemo." He's drained from all of his therapy and all of the help and he's DONE. "But counting down the days to go." He's getting help - "chemo" - BUT still counting down the days to go. It's not helping him and he's ready for death, just counting down the days. "It just ain't living." This is his reasoning, his explanation, as to why he's killing himself. Because it just isn't living. "And I just hope you know that if you say, goodbye today, I'd ask you to be true." He's saying this to his friends and family; that if they're going to say their final goodbyes, they need to be true. As in, not mope about him going and honestly say goodbye. He needs them to accept it and let him die, because the hardest part is leaving them.