。
She hates speaking in public.
Literal
She [topic-は] in-public [at-で] talk [object-を] doing [nominalizer-の] [object-を] detests.
The nominalizer の turns the entire clause 人前で話をする ('to speak in public') into a noun phrase, which then takes を as the object of 嫌う. Note the verb 嫌う (きらう) — it's the verb form meaning 'to detest, to loathe,' distinct from the na-adjective 嫌い (きらい, 'disliked'). Both express dislike, but 嫌う is more emphatic and literary, while 嫌い is the everyday conversational choice.