Where are the cheap hotels located in Sydney ?

Given your limited budget of $50 a night, you could stay at a backpackers. There are some backpackers to be found in the innercity including YHA's (Railway square, central & the rocks), wakeup (sydney central) http://www.wakeup.com.au/ & the George St backpackers http://www.790ongeorge.com.au/ If you want to stay at a "hotel", try The George Hotel. If you are prepared to increase your budget a bit, according to their website $67 is the daily cost of a single room, with shared facilities. http://www.thegeorge.com.au/budget.htm And a quick look at the standby rates, states that currently there is weekly rate available for a single room of $308, which seems like good value.

Newtown is 'the new Cross'. It's a happening place with two excellent backpackers: http://www.australiansunriselodge.com/index.html#rates http://www.billabonggardens.com.au/newtown.html An easy $10 taxi or even cheaper on the train into the city (or walk, I've done it)

You could try using a website such as wotif.com or expedia. Those would bring something up. But for $50 you would be looking at a pretty dodgy room, in a backpacker place or without your own bathroom.