The meaning depends a lot on the context. I can't tell the exact meaning without knowing what he was talking about. It could be 'I can't do better than that' - e.g. "That's a great score, I'm sure I can't beat it". If he was talking about the depression, it probably means "I can't get over it", or "I can't get cured of it". You can use it for an illness that you can't cure. "Despite all I have done, this cancer is spreading. I can't beat it", "I've tried every product on the market, but this fungus is tenacious. I can't beat it".

Having watched Manchester by the Sea, I can verify that the man (Lee) was talking about his depression. In English speaking countries, depression is seen as a thing to fight against. People view it as a negative mentality that must be conquered. Therefore, it's common to say "I'm fighting depression" or "He's fighting depression." So when Lee says "I can't beat it," he's referring to his depression. No matter how much he fights it, by spending time with friends or going on boat rides or drinking beer, he still can't beat it.

to beat = to defeat. This word is used often for sports and other competitions but it can be used for other things. To say that you can't beat something is to say that it cannot be defeated by you because it is stronger or superior. In this case, the man feels like his depression is so strong that he will be depressed forever.