A while ago I arranged to have the electrician who works for the landlord come by to replace the phone outlet and any wiring inside the house because of our problems with the internet. Unfortunately, by the time I'd gotten around to ringing him, he was on a long term contract down in Melbourne.
He finally made time to come by while I was at work last Wednesday, and it turns out that there's almost no customer wiring inside the house - the demarcation point (I'm not sure what they call it in Australia) is directly behind the socket. He replaced everything with brand new stuff, and for cheaper than I was expecting too (then again I thought it'd require a visit to the attic).
The dropouts are mostly gone - it lost sync once about 24 hours after his visit, and at the time of writing it's been up for almost a hundred hours since.
The noise margin and other associated numbers are still much worse than I'd expect being so close to the exchange (we're probably no more than a kilometer or so from the exchange in a straight line, but I don't know the loop length), but at least it's reliable thus far.
We'll see how it goes, and if we still have problems, now that I know I won't be held financially liable, I'll report it to Exetel.