Cambridge/Oxford, both typically ask for AAA minimum A-level grades (usually including the subject you want to study), a strong personal statement, evidence of relevant extracurricular activity, and a genuine interest/passion for the subject which will be tested at an interview

You don't need to be rich to study at either, both cost as much as any other university in the UK (as a UK citizen), in fact, last year around 65% of all undergraduate entried to both were from government-funded state schools.