Can you read at all? Is there any sort of written material that you can effortlessly concentrate on for half an hour or more, such as textbooks, class notes, newspaper articles, tabloids, blogs, facebook posts, or forum threads? If the answer is no, you may have some cognitive disorder and should seek professional help.
If the answer is yes, evaluate how you motivate yourself to read the things that you do read. For me, it's a matter of being bored and wanting to kill time by hearing something interesting and thought stimulating. Then the only thing that remains is to make a decision to read War and Peace instead of Stack Overflow. Keeping the book on your desk where you can reach it to reduce transaction costs helps.
One thing I've noticed, as someone who oscillates between avid reading and complete apathy, is the illusion of non-commitment. It is easy to think that (e.g.) Stack Overflow is a better use of time than a book, because you would like to spend very little time (like 5 minutes) on your leisure reading, and you obviously cannot achieve much by reading a book for 5 minutes but the opposite is apparently true for more bite-sized material.
This is, of course, nonsense. If there are people who have the discipline to decide to "check out my newsfeed for a few minutes" and actually not have it grow into at least half an hour, I haven't met them. The true solution to your problem is to let go of this fallacy and start being realistic about what sort of commitment the different activities entail.
It is a bit less challenging to remedy the problem superficially, without addressing the root issue: In times like long train journeys, or waiting in line at some government office, you know you are wasting your time anyway, so it's not hard to convince yourself that you might as well read a book, since what else are you going to do instead? This is quite effective, and I have finished many books during long commutes by always keeping a book on me. So, in essence, force yourself into a situation where you are bored for extended periods of time, and have nothing to do except read a book (and make sure you have a book with you at such situations).
Unfortunately, nowadays this is harder - chances are that you have a smartphone, which will allow you to surf the web or browse facebook, and thereby compete with reading time. If you can't effectively pre-empt this competition, you will not want to read (because of the greater perceived -but not actual!- commitment for reading) instead of browsing on your phone. I do not know of an effective solution in that case.
Lastly, a point on reading material: You should be careful how you choose your books; there is a difference between books that you want to read and books that you want to have read. Some classics are just hopelessly boring to some people, and some people just like reading trashy fiction. Of course, it's best to try not to read nothing but garbage, but that is up to a point: You can only force yourself to read boring books that you think every intellectual such as yourself ought to have read so many times before the whole act of reading becomes insipid to you. Good luck getting a habit of reading then!
In practice, I find that the most effortless reading is from spontaneous decisions - just walk down a random part of your library or bookstore, and pick up books at random (maybe the title sounded clever, or the cover picture struck your fancy). Read a page or two, just enough to figure out what the book is about. You may discover that such an exercise often ends with you on page 50 and not wanting to stop, at which point you check the book out and finish it.