The short answer is no, although machines do make the job considerably easier, as it is essential to beat air into the mixture while it is freezing to break up the ice crystals and create a rich, smooth texture.
To make ice-cream by hand, pour the mixture into a metal or plastic container and put it into the freezer for about 1.5 hours. When it has started to freeze around the edges, you can either whizz it in a food processor for 20 seconds, or tip it into a bowl and beat vigorously by hand or with an electric whisk.
Return the slushy ice-cream to the container and pop it back into the freezer, and repeat the process another two or three times until you achieve the right consistency.