I (21 years old) moved to the city only a few months ago, and I am now living in Hull (Gatineau), where I found a 1 bedroom apartment that costs just $650/month. It even has a private backyard and a parking space. It's about a 5 minute drive from Parliament Hill.

There are some pretty good apartments on the Quebec side if you are patient enough. I was in a shared house in Ottawa for 2 months before getting this place, but that was only because I didn't have a full-time job yet, so no landlord was willing to accept me as a tenant.

Also, if you stay on the Quebec side you at least won't have to switch everything (health card, license, etc) over to the Ontario side (I have to switch everything over from Ontario to Quebec and it is a pain in the ass).