What matters here is your last statement - the fact you're booking it as a single ticket.
When you book a single ticket - regardless of how many airlines you are flying - your ticket is issued by a single airline. In this case that would probably be LAN Chile (as they are the first airline you fly), although it's also possible that it would be Air Canada (as they are the longest flight). For simplicity, lets presume that it's LAN Chile.
When you book your ticket, LAN Chile is basically committing to get you from Auckland to Vancouver, as per the rules in their "Conditions Of Carriage" (CoC) which is basically the terms and conditions that go along with your booking. In short, as long as you don't do anything to break the terms (eg, failing to turn up for your flight), then they have to get you from your starting point to your destination - regardless of issues like delayed flights on ANY carrier.
In your case, that means that if your flight out of Auckland is delayed/cancelled/etc, LAN Chile has to make good on getting you to YVR. That might mean anything from re-routing you (eg, putting you on a AC flight AKL-SFO-YVR), putting you on a different airline (eg, SYD-YVR on United via LAX), or simply putting you onto the AC flight the next day. Obviously exactly what they would do would depend upon availability, how delayed you were, etc.
In the event that you end up having to overnight, some airlines will put you up in a hotel, whilst others will not. Some will do so if the delay is their fault (broken plane), but not if it's outside of their control (bad weather). This is where travel insurance can come in, as most policies would cover the cost of your hotel if you're delayed.
If you were booking these two flights as different tickets then it's a very different story. LAN Chile has to get you to SYD, but if they get you there too late to make your next flight then that's just "too bad". Air Canada would then claim (correctly) that you broke the terms of the CoC on their ticket by not arriving at the airport in time, and thus would normally cancel your ticket. Depending on the airline, they may choose to show you charity and accommodate you on a later flight (eg, next day), but they have no responsibility to do so, and if the next days flight is full, you're pretty much out of luck! In this case most (but not all!) travel insurance policies would cover you, although some may only cover you to the price you originally paid for the missed flight, even though the last-minute replacement flight could cost significantly more - so read the fine print on the policy!
Sydney is a relatively easy airport to transit, and although the lines to re-clear security can be a little long when the LAN Chile flight is due to land, they will normally have cleared out an hour or so later - ie, by the time you'd be coming through if that flight was delayed it would be fast, and you could probably go gate-to-gate in no more than 10-15 minutes. 3 hours gives you at least 2 hours worth of buffer, so unless you expect the LAN Chile flight to be more than 2 hours late you're good to go!