Do Possums live in nests

Yes and no. True possums, which are those native to Australia (and introduced to New Zealand) live in hollows in trees, and some species line their hollows with shredded bark and grass. This could be regarded as a "nest".

Possums which inhabit suburbia, such as the common brushtail, often live in the roof spaces of houses, or any other available space such as chimney and even doghouses.