EU/EEA members (plus Iceland, Liechtenstein, Norway, and Switzerland nationals) can work and live in Greece without a permit. They are only required to register with the authorities in Greece. Additional information can be found here.
Aupairs from outside of the EU must contact their local Greece Embassy to see if they qualify for a work permit.
Resident permits are very difficult to obtain. They require Government approval, a large deposit, by aupair or employer. Very few are approved. Greece has the lowest approval rate of any resident permit. You need to first obtain a visa to enter Greece then apply for a permit within 30 days of your arrival if you intend to work there longer than 90 days. The process from start to finish, could take up to a year.