Eben,
Not wanting to disillusion you but somehow I reckon you're near approaching mission impossible (there are going to be difficulties), not with your move because that is fine but just seeking furnished accomodation up to $500/w.
I don't live anywhere near Sydney but know the city reasonably well though I had to re confirm my belief where Pyrmont was only because I was sure there is probably limited accomodation about there.
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Sydney Map - Google Maps if you haven't already.
On top of that the Sydney realestate/rental market is as tight as a drum, well tighter in fact and I reckon for family furnished accomodation you would have to look at two options (or a few) but just of the top of my head:
. you will have to look for somewhere very outer suburbanish , perhaps even country town (and little point in looking coastal as there is always a premium attached, especially with summer coming) and your wife looking at a rather longish commuting hours.
So using the mapping and following railway lines (south, southwest and west) put a few suburb names into
http://www.australiaforum.com/housing/www.domain.com.au and see what you find.
You may have to commit to a minimum three months lease if something is available and an associated approach could be to google up a suburb/town name and look for acommercial link that could list real estate agencies, Ray White, The Professionals, Century 21, LG Hooker being a few that spring to mind and then make a direct contact for especially in outer areas they may have been contacted by someone seeking house sitters or know of an older style farmhouse or something that an owner is keen to let, and even if it is unfurnished, that could partly be a bonus for if say $200/w cheaper, a $1000 spent at secondhand and community stores could set you up.
. second option, a variation on the above in that to save your wife all that travelling, maybe she considers staying in town during week while the mob is out at the country retreat! - hostelling if she can hack that being best approach and she may have to skip working and take a Xmas break for most hostels will already be booked out.
. third option is the whole family kind of hostels it - most have four bed dorms, some with ensuites and one I stayed at in Sydney and have mentioned on here to others is Alfred Park (on
BUG - the Backpackers' Ultimate guide to budget travel in Australia) for part of it is an old motel - nothing too flash but clean and comfortable and the sort of place you might be able to do a deal with management to take one of the motel room dorms so at least you have a fridge, tele and ensuite (and a big park on opposite side of road)
And Central Railway Station is on other side of the park and there is a light train/tram service from there down to Pyrmont.
. 2 & 3 would also give considerable savings on transport.
Sydney at least has plenty of attractions and with warmer weather and daylight saving you can be out and about exploring so it is not as if you would have to be cooped up too much.
As to deal, it could be you could negotiate a rate getting down near that $500/w for a 8-10 week stretch and if Xmas bookings are a problem, maybe you have to get yourself some camping gear and head off down or up the coast a bit for a Xmas camping holiday.
Another place that could be worth considering to do something similar is
Backpacker accommodation Sydney, Sydney Terraces, budget, cheap, hostel, taking two units and sharing facilities.
But it is going to be more a case of what is available and booking it quick rather than being able to pick and choose, and we have a few posters in the RE biz it seems and so you might even get some PM offers, but they could also be involved more in investments and sales rather than rentals.
Good luck and do let us know what you can arrange for it will be good info for others like minded, and also ask re info/insight on particular locations if desired.
www.gumtree.com.au is also a site where you might find something on offer.