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She cleaned the house without anyone's help.
Literal
She [topic-は] anyone [genitive-の] hand-[also-も] without-borrowing house [object-を] cleaned.
手を借りる ('to borrow someone's hand,' i.e. to ask for help) is a fixed idiom; here it's negated and turned into a 'without' phrase via the literary negative 借りず (the classical ~ず, equivalent to modern 借りないで). The 誰の手も~ない pattern combines a possessor question word + も + negative — 'no one's hand at all.' Together this gives the polished 'without anyone's help.' The classical-flavored ~ず gives the sentence a slightly literary feel; in casual speech you'd hear ~ないで instead.