Huset is a bit of a gimmick - you'll really only go here once and it's predominantly for the tourists, not the locals.
They offer a changing set menu which features local fare like reindeer or sometimes seal or whale.
The restaurant is very cosy and the presentation and service of the food is very good.
I wouldn't quite say it's "fine dining", more of an upmarket fancy restaurant but the experience is not exorbitant like most scandinavian fine dining places. DInner for two was around $280 USD with wine for one three course meal and one five course meal.
Worth doing once!