Norway have many cabins available all over the land.
DNT (The Norwegian hiking association) have many cabins where you can spend the night and marked routes you can follow. Some cabins even have hosts and serve food. Read more about DNT here.
If you want to be guaranteed to be completely alone in a cabin, and rent it for yourself, there are some hidden gems all over the country. The fishing and hunting associations have many cabins you can rent, as do other providers like local clubs. These aren’t as easy to find as the others. Look at this cabin for an example of a real gem. The prices are pretty cheap as well, even by norwegian standards.
Go to inatur (press 'hytter') to see many available cabins.