According to Paul Rustchynsky, Game Director of Driveclub, the game will take 17 GB of storage. After downloading the first 3.5 GB of data, you can begin playing, while the rest of it continues to download in the background.
Driveclub will be in addition to you regular PS Plus allocation. There’s loads of content included. Do try it out – I think you’ll have a great time with it, and at no additional cost to your wallet.