。
Does she have any hobbies?
Literal
She [topic-は] some hobby [object-を] is-holding [question-か]?
持っている is the resulting-state form of 持つ (to hold/have) and is the standard way to express possession of objects, qualities, or hobbies in Japanese — bare 持つ rarely appears for ongoing possession. 何か here is the indefinite 'some' (built from 何 + か), modifying 趣味. The question is addressed to a listener about a third party, a common shape in casual social inquiry.