In Århus, we stayed at this hostel in a park north of downtown. We'd heard that Danish hostels were pretty nice and we weren't disappointed. The rooms are small and spartan, but they're also clean, cheap (by Danish standards) and include a great breakfast.
When I say "the rooms are small," I mean it.
Den Gamle By ("The Old Town") is a living museum in the same vein as Old Sturbridge Village or Greenfield Village. It's full of old buildings from the 16th through 19th centuries moved here from all over Denmark. When it opened in 1914, it was the first museum of its kind.