Yes, it is possible to live on the moon. In fact, it's possible to live possibly anywhere where the laws of Physics known to us still hold. It's a matter of having the right technology with you. If people where to live on the moon they'd have to live in underground constructions most probably - this is because of the barrenness of the Lunar surface and the extreme temperatures on it. The moon has no atmosphere hence people would have to live on an Oxygen supply carried from the Earth. They'd also have to have a food and water supply that would last them for as long as necessary. If people were to live on the moon they'd rely on solar electricity for power generation. Protective wear, of course, would be necessary when stepping outdoors- hence space suits would also be necessary. Thus life on the moon is quiet possible