I am afraid BF won't marry me

I think you run the risk of pushing him away if you put pressure on.  You are vey uoung to be dealing with this damn disease and I think some counselling with a Counsellor who Works with cáncer patients would really help you. 

Studies have shown that when people 'decide' on something, they tend to be more satisifed with their choice, than if they still have options. I just listened to an NPR podcast on this and the scientist, who had a lovely live in for years decided to propose after realizing the study results. Marriage has more perks than just a 'government paper'. Here's the podcast on making decisions: https://www.npr.org/podcasts/510308/hidden-brain

That said, you can't put your life on hold waiting for him to be ready. You are in your 30s and it's still to follow your dreams and find your niche. Maybe your dependency on him is what is keeping him ambivalent? Perhaps let him know that if he's not ready to settle down, you are and you'll need to move on and keep looking for someone ready as well... That may help him decide it's time and if not, move on and find someone more mature.

". .. he says he is not ready yet and he needs time. It makes me feel sad, jealous, and even angry when I hear a friend or a coworker just got engaged, went to Europe for honeymoon, etc, I just want to be like them, healthy and happy! I was only 29 when I got this damn disease, I just want to the rest my life can be a little easier! Please give me some advice how to get out of this downward emotional spiral?"

Stephi, it really, really sucks that you got BC at age 29 and your anger is justified, it is so unfair.

As for the BF, and arguments by some that marriage is "just a piece of paper", I strongly disagree, especially if people are still young ( some older people do not want to marry because their social security payments go down).
I am 63 and came of age when free love and living together without marriage became accepted by society, even considered hip. It was great for men because they could get free sex on demand without having to commit to a woman, but could still easily leave for someone more interesting or more attractive without legal hassles, alimony payments, or other separation headaches. Not wearing a wedding ring also signals to other women that a guy is probably still available, as there are many women, yourself probably included, who would not dream of messing with a married man but if a couple is just living together, well, there is understandable hope he could just leave, or wants to leave.
Women were often left holding the bag emotionally when this would happen.

Call me a cynic, but I have been around the relationship block too many times to buy the line your BF is giving you. I would absolutely get counseling, ( and possibly a psychiatrist for depression medications), as others have suggested, and consider asking the BF directly what exactly he is not ready for or needs more time for, e.g., is it the uncertainty of life with a woman with BC that puts him off, worry you will want children, worry he will have to support you, that he is still looking around, or what exactly. The answer may be that he is not just not ready to commit now, but possibly never will be, despite being there for you for the last year. He may still be there but not want to marry you because he has not yet found a suitable replacement, so he is keeping options open by not proudly putting on that ring for all to see.
You may still want to stay with him even if he is uncommitted, but it should not be because you think women with BC have to be "alone." They do not, as others said. I know men who have married women with a history of BC, good solid men, and I would not cut yourself short.
I wish you the best, which you deserve.