Yes, yes you can. I've rented places like that in Fiji, Samoa, Costa Rica, and Bora Bora for week+ time period. Hotels wont really cut you weekly deals, but if you look for backpackers they will offer weekly rates.
Also in Bora Bora Tahiti, you can also do homestays which aren't as flash as 5 and 7 star hotels, bt it's cheaper and if the family likes you you can stay as long as you want, and you get a lot of culture.
Samoa has a sort of "hut" system like the States have. It's great. A family manages the hut on their property which all are right on the beach. You pay them 10 bucks a night. You don't have AC TV and sometimes electricity, but what you exchange for it much better.
Costa Rica also has a lot of homestay like accomodation.
Unless you are purely looking for a hotel, you can pay a hotel any amount of money.
Our place in Fiji was the best though. It as a small backpackers on a private island. 6 of us stayed in a hut that was 10 metres from the beach. We had 7 acres to roam, and the total amount of people staying at the resort wasn't more then 30. Sunset happy hour every day, cheap cheap really great food. I got away with staying 2 weeks paying no more then $150 for accommodation and $200 on food and beer.