Not how did they preserve it, like with smoking and drying, but how did they keep it from animals? I’ve been camping enough to know that bears and chipmunks will find a way to get into any food supply that isn’t inside a modern invention (concrete bear box, bear proof containers, etc). How did they keep their food safe without modern bear proof technology?
Native Americans is a very broad term as there are hundreds of Tribes (or bands in Canada) that would have historically covered the entire lower 48 as well as Alaska, Hawaii and other parts of the US. I’m sure that other people can weigh in on their particular region, but I can focus in on the Haida people who lived (and continue to live) on the Northwest Coast of British Columbia in Canada as well as the very southern part of the Alaska panhandle. One thing to hone in on in terms of OP’s question is that food (for bears as well as people) is incredibly abundant in this part of the world. There are numerous species of plants, fish, invertebrates, and animals that would have provided ample food for bears. The notion of “bear proofing” is a very modern idea that really comes from camping or backcountry travel. The Haida have zero oral history that I’m aware of where bears are considered pests or that food would need to be safeguarded from them. The pre-contact Haida lived in relatively large villages of several hundred people which would have several large longhouses arranged so that they face the ocean. The longhouses are quite large and square so they would have had ample space for storage and are very cozy and dry due to a open fire hearth in the dugout centre. Longhouses were constructed with red cedar which is naturally decay resistant and would have provided a lovely smell. The entrances to the longhouses were very small and the Haida generally had a small round opening in the front which was the main door. This door would have been impossible for a bear to enter as it was designed to force an intruder to bend down as they entered. This unique door safeguarded the Haida that anyone attacking their home would have to enter one at a time.
The Haida stored food in bentwood boxes. These were made of planks of cedar that were steamed and bent to provide a very sturdy container for dried or smoked salmon, dried berries, oolichan grease or other food that could be stored. All of these foods which would have been (and still are) staple foods for Haida.