North Korea does have Internet access - for a few thousand party elites. The few thousand other people who have any kind of computer access can log on to Kwangmyong, a sanitized, heavily restricted Intranet run by the government. Any other North Koreans are S.O.L. - but they are probably more concerned about other, more pressing needs anyway.
Kwangmyong, which is Korean for "bright star," is North Korea's officially sanctioned intranet. It looks sort of like the internet circa 1994; many users even access it with old-school dial-up or computer labs. It is a closed network that runs on pirated Japanese versions of Microsoft software and looks sort of like the real internet but isn't. Rather, it runs rudimentary email and browser tools that are restricted to a hand-picked collection of "sites" that have been copied over and censored from the real internet.
This network is accessible by the handful of computer labs at major North Korean government offices, universities, and a small number of cyber cafes in major cities. (Internal travel is forbidden without permission in North Korea, so most citizens never see Pyongyang or can visit its cyber cafes.) But you need a computer to access it, and that's only possible with official permission.