There are camping facilities close to Canberra at EPIC (Exhibition Park in Canberra) see http://www.epic.act.gov.au/camping.asp
Depending on what you class as close to being close to Canberra City there are also facilities at Cotter Dam, the Brindabella Ranges and a bit further out Wee Jasper.
I work in Canberra City and live in Yass so I have never camped in EPIC but have driven past the grounds any number of times going to the events there. It looks average - maybe OK for a couple of nights if you are overnighting and want to head into town. There is little there as it is mainly a campground for people exhibiting at events and the various shows.
Cotter Dam is about 40 minutes from the Civic, and Wee Jasper might be a bit far it would be over an hour and a half.
If you are looking for a CP you could try Queanbeyan.