I in general think that answer to this question differs for scholars and for general public. In order to do scholarship, which for me are activities of writing, reading, transcribing, researching, encoding and formatting texts, it takes hours of uninterrupted time. To be productive, I resort to several practices which in our era of information overload seem almost like monkish or meditative, including the following:
1) No smartphone. Never bought one, originally because expensive, later because decided it was a time sink and source of interruptions. Not having one has mostly eliminated the sleep problem.
2) Limit social network usage to minimum (no Facebook, never signed up; did use Twitter several years, now mostly in hiatus)
3) Anti-distraction software on desktop (Macintosh SelfControl, which blocks access to time-wasting web sites)
4) No 24-hour TV news (no cable)
5) Over time cultivated and developed preference for austere and regimented software interaction, plain text with RegEx manipulation, LaTeX, well-formedness and validity in XML, citation keys with Zotero, etc.
Sustained scholarly engagement demands immense rigor. And I usually fall short of what I would hope, but the choices and habits above allow me to not feel particularly overwhelmed by digital culture. These obviously cannot be adopted by everyone. I'm past middle-age and long-time partnered, so I have no urgency to keep up with youth culture or to date. A public or activist scholarship would likely be unable to restrict itself to such methods. And I wonder how a scholar of gaming culture can resist the allure of just playing games. But over time I've satisfied myself that being aware of the daily political news or sports or cultural events lacks the urgency that I once felt toward it.
I was influenced by listening to a scholar on NPR several years ago, perhaps on FreshAir. An eminent, productive scholar, she described limiting her access to the news so she could concentrate on her work One day her mother (the scholar's mother) wanted to talk about the fall of the Berlin Wall. And the scholar had to explain to her mother that she did not know that the Berlin Wall had "fallen" several days before--because she had restricted her consumption of the news. Her mother said something like, "I don't want to talk to anyone who does not even know about the fall of the Berlin wall," and hung up the phone. The scholar, at first in some consternation because her mother had hung up on her, resigned herself to her choice, though she protested to the radio audience that she usually learned about major events within a week or two (i.e., her mother had told her), and remained sufficiently informed about the present. I wish I could recall the scholar's name, but her attitude seemed immensely sane to me. And I cut back on news consumption after hearing that interview.
I don't think it's easy to draw line, but I do think it's possible and necessary to cultivate habits of information interaction that reduce the feeling of being overwhelmed, for scholars and the general public. In my case, to concentrate on scholarship, it feels like I have to draw the lines with a degree of rigor that to the general public and my students seems excessive.
Snowflakes are falling in abundance outside my window. I paused while drafting this answer to watch them.
Best, Wesley
