I'm not seeing anybody else suggest what seems to me to be the obvious solution: opt out of twitter (which seems like a self-abuse platform) and filter your email. Part of why communication works is because we can't hear everybody's opinion on everything. People earn the right to our ear through networking and earning our company. Twitter undoing this is not progress.
So... the answer, for me at least, is to turn them off. I deleted my Facebook and Twitter accounts. I don't watch, read, or listen to news.
"How then do you stay informed?" you may be asking. Well, I didn't delete my family or friends and they will often bring up topics that are of interest to me. It's about to snow a lot where I live. I heard about it first from my friends and then made the conscious decision to read more about it online so I could know and be prepared.
Sure, I also get the occasional bullshit political vitriol but they're quickly realizing how ineffective that is with me because I either unceremoniously change the subject or walk away from them when they start that nonsense.