The way Goldie Hawn moves is incredibly natural. I literally find it hard to believe that they were using cables to hold her in the air, especially on the parts where she slides back through the floor.
It is a technique known as wire work, often used in martial arts films—see Wire fu.
A wire is attached to a harness under the bodice of her dress is threaded through a pulley on a crane. Typically a man is on the other end of the line, pulling down to raise her a little, or two men to raise her a lot. The smooth horizontal movements are made by extending/retracting a telescopic crane, or for limited distances, a stationary crane can be rotated. You can see some stunt performers training in this video.
In post production, the wire is removed through a technique known as rotoscoping.