I'm currently living in the family home in Sydney but the family is very dysfunctional. I've been saving up for a while and should be able to move out by the end of January 2016. I would love to major in economics and finance, and fill up the rest of my course with econometrics and maths/statistics units. I'm not too sure what I want to be doing as a career but I definitely want to open as many doors as possible, maximise my networking opportunities and get relevant practical experience whilst at university. One day I want to go into the baking and finance sector, another day I want to go into consulting and on another day, I want to work for the treasury - clearly, it's better just to keep as many doors open as possible! I'm not sure if it's best for me to stay in Sydney and go to UNSW/USYD or move to Melbourne and study at Monash/UniMELB. Please let me know what you think! 
Well, you're definitely right about building a massive network if you want to keep your doors open. One advantage to moving, which isn't necessarily due to your choice of degree, is the networking opportunities that you'll force yourself into. For example, if you stay on campus at Melbourne, you'd be forced into social situations that you could easily avoid if you were just commuting to USyd from home.
Either way, if your household is dysfunctional enough (very sorry to hear it, by the way

) for you to plan extensively to leave it, then you should probs move out (especially if your family is the type of dysfunctional that will still assist you financially). Then it's a question of whether to move to Sydney or Melbourne.
As for advantages of Melb vs. Syd... there probably aren't that many. All are phenomenal unis to graduate from.