This question is a bit tricky because I am unsure to which part of the book you are referring. I am going to venture a guess, however, that you are asking about Julie/Miyax when she plans to leave her arranged marriage. In that case, she fears a few things: that she will be sent back to her Aunt Martha, that she will continue to be raped, and/or that she will lose her way on the tundra. (Unfortunately, the third thing actually happens.)
First, Julie is afraid she will be sent back to her Aunt Martha. Julie is forced to live with her stern aunt in order to go to school. This separates Julie from her father and the seal camp where she is able to live in harmony with nature. The only way she can escape from Aunt Martha is by agreeing to the arranged marriage to Daniel.
Next, Julie is afraid of Daniel continually trying to rape her. Being a mentally ill young man, Daniel is not always in control of his own actions. He has tried once to rape Julie. There is no doubt he will try again.
Finally, Julie realizes the reality of being a lone person attempting to traverse the Alaskan tundra. She desires to make her way to San Francisco, but knows she has a slim chance of finding her way. Julie, then, fears death in this instance: death in the Alaskan wilderness.
In conclusion, the third fear partially comes true. Luckily, Julie comes upon the wolves, who accept her into their pack. Further, Julie eventually finds her father once again.