Here, haar is a possessive determiner, meaning her.
It belongs with eie storie:
So in this sentence, haar does not mean the object pronoun her. It means her in the sense of possession.
Afrikaans verbs are much simpler than English verbs in the present tense. The verb usually stays the same for all persons:
So unlike English, Afrikaans does not usually add a special -s form for he/she/it.
It can often mean either, depending on context.
So Sy skryf haar eie storie could mean:
If Afrikaans wants to make the ongoing action extra clear, it can use a longer form such as:
Sy is besig om haar eie storie te skryf.
But the short sentence is completely natural.
Yes. This sentence uses the basic subject–verb–object order:
So it works very much like English: She writes her own story.
That said, Afrikaans word order changes in some other sentence types, especially after certain sentence openings or in subordinate clauses.
Yes. Storie is very close to English story, both in form and meaning.
That is common in Afrikaans: many words are easy for English speakers to recognize, even if the spelling is slightly different.
Also: