Doing nothing is the wrong concept. You never do nothing, because even when your body is still your mind is churning and processing information.
I have a strong dislike against "wasting time." I don't like myself when I spend time on nonsense. And so I fill all of my day with "constructive things." My walk to work is filled with podcasts, the time waiting for the food to bake filled with news articles. While eating I entertain myself with shows or Ted talks or whatnot.
The best decision I made in the last weeks was to stop most of that.
Aristotle recommended to take walks - especially while discussing with another person. And now, walking to work with just my mind and the scenery and passing people as company I feel more relaxed. I feel serene. I learn to understand myself better, just the way a meditation clears my mind.
I mentally plan my evening or reflect on the day - conflicts with the boss, troubles, things I achieved, things I learned. I finally notice the food I'm eating.
The list goes on. I'm not going to stop consuming information and I'm not going to stop using podcasts on some long walks - but I live more consciously, more aware, more relaxed. It's small changes and suddenly I'm happier and can handle stress better.
I think we all tend to drown our minds - emotions, thoughts, worries, little wins, conversations we had or want to have and much more - we drown all of it in manufactured emotions (reddit, games, tv, ...) and interesting, and valuable, but ultimately unnecessary information.
When you say "doing nothing" you confuse something. You are doing things all the time, your brain never takes a break. But when you "do nothing" you finally allow your brain to breathe and process all the things it needs and wants to process. I think all these modern diseases - sleeping problems, stress, depression, distractability, even obesity,... - they have a lot to do with the fact that we don't allow our brains anymore to breathe. We bombard them with stuff - either information or, worse, emotion - and in order to handle this stuff other important tasks - housekeeping tasks such as consolidating memories, reflecting about one's feelings and health and happiness, planning healthy food, considering how to bring up that issue with the boss - are drowned in a sea of emotion and information. They are drowned in a wonderful wealth of "stuff to process" that ultimately prevents our brains from ensuring their own - our - mental and physical health.
We are indoctrinated with an idea that time needs to be "spent". That's why you wonder what people do when they don't do all the things you do. I tell you what: they engage with others and, more importantly, with themselves. They learn who they are and what they value. Without any effort their minds plan the future and consolidate memories of the past.
That, I think, means to be truly alive. "The unexamined life is not worth living," said Socrates. The modern version is maybe this:
The person that lives solely in emotions and information from the outside, the person that never pulls itself out of this messy reality and gives itself over to a mental spa, a time of healing and processing, a time of reflecting, feeling, thinking, seeing, worrying, planning, smiling, that person doesn't live.
Take a walk. Leave the iPod and your phone at home. Find some trees or a place with a nice view. It's even okay if you just lie down on the couch or stand in the shower or sit at your desk, with your eyes looking past the screen. Just be you, for a moment. And then watch, carefully, without judgement, all those things that happen in your mind while you "do nothing."