Primrose Park in Cremorne is a great spot for kicking a soccer ball around. In fact, it's actually used for kids soccer games on the weekend, so the ovals are always maintained. It's a dog friendly park, but their are two huge ovals and plenty of grassy hills, so you won't be fighting for space.
http://sydneyfun.com.au/primrose-park-cremorne/
Otherwise, St Leonard's Park in Crows Nest is another good option, although the oval area can get flooded after heavy rain. If you'd prefer somewhere with less people, I'd try out Rawson Park in Mosman.