Yes, as long as you're not a journalist or a professional photographer, you can travel there. And it's an amazing experience.
It is true that the country has problems, and you should never forget this while there. You could get your guides in trouble by doing anything stupid - and this starts with really small things already. For instance, our guide asked us not to fold any North Korean newspapers (because you might fold a picture of one of the Kims by accident) and not to discard them in the waste-paper basket. If we didn't want them any more, we should give them to our guides and they would take care of it.
Read some books about North Korea before going there, to give you an idea what you're dealing with, for instance "Nothing to Envy" by Barbara Demick. And do NOT take this book with you. Your guide book about North Korea should stay at home as well, and the same goes for any religious writings. Just take an inoffensive novel and you'll be fine.
There are several tour operators that take tourists to North Korea. I've been with Koryo Tours last year, and I've been satisfied. Your travel group will be accompanied by two or three North Korean guides and a driver at all times outside the hotel, and you can't leave the hotel unless at least one of your guides is with you. These guides are the reason you should not just try to get the best price on this tour. You should try to get the best guides. So look for recommendations online about the tour company you are planning to travel with.
These tours are designed to leave you with a good impression of the country. So normally, you should have a good time. Keep any critical thoughts to yourself - your guides don't need you to tell them that things aren't perfect. They know.