There's no technical requirements proscribed by the Constitution. In fact, the Constitution only mentions 1 judge, the Chief Justice. Every other federal judgeship, including the number of other Supreme Court judges is determined by Congress. They set the requirements, but essentially anyone appointed by the President and confirmed by the Senate can be a Justice. There are, however, some informal near-requirements.
Law school for starters. 25 of the 59 Justices that graduated Law School (it wasn't always common among lawyers, they used to apprentice instead) graduated from Harvard or Yale (including 11 of the current 12 Supreme Court justices, Stevens went to Northwestern, that's part of why Harriet Miers and her Southern Methodist University Law Degree got laughed out of the Senate before her confirmation hearings even began.)
Most Justices also come from experience sitting on one of the 11 US Circuit Courts of Appeals. I don't know the statistics, but I think all 12 current justices came from that background. Few Justices reach the Supreme Court without being a judge first. Former President Taft is a notable exception. I suppose Obama could nominate Bill Clinton if he so chooses, but I find that highly unlikely. Supposedly, Obama did consider nominating the Solicitor General. Justice Thurgood Marshall was the Solicitor General when he was appointed, but he had previously been a federal judge as well.
Many Supreme Court Justices early in their legal careers find themselves clerking for other notable judges, including then current Supreme Court Justices. Successful legal and political careers are also helpful, which is generally true for any judge. Becoming a judge isn't too different from getting any other political job. Some are elected, some are appointed, either way it's very political. The ones who reach the Circuit Courts are all very good at politics. The ones who reach the Supreme Court are even better at it.
Generally, they need to have a good record without too many crazy decisions, however that's not always that important, so long as the President agrees with their judicial philosophy. It's very rare for the Senate not to confirm the President's selection so long as they meet the basic qualifications above. Even Clarence Thomas got confirmed and he was accused of sexual harassment. Judge Robert Bork was famously not confirmed by the Senate. To "bork" is not term for failing to confirmed a Supreme Court Justice. Bush withdrew Harriet Miers from consideration on fear that she would have been borked. It is VERY rarely done, however.