Firstly, does Zurich airport allow you to sleep over night in the departure area? (My layover is the middle of winter so I don't want to be sleeping outside.)
If not, what is my cheapest option for off site accommodation?
If you don't have access to frequent traveller lounges, and you don't want to sleep in a chair, you can buy a ticket for one of the day rooms offered by the airport, which will cost you CHF84.
sleepinginairports.com also recommends the Starbucks at the airport, if you don't really need a bed.
In the city, which is easily reached by public transport, the youth hostel (40 min from the airport) is a reasonably priced option; how much you pay depends on how many people you're willing to share your room with.
The Zurich airport does not close overnight. Even though there are no flights overnight, you can generally stay there and sleep on the chairs. Even though it is not really comfortable it is more or less safe. It could be that you will be woken up by security, but they will only very rarely tell you that you have to move on. If they do so, just go around the corner to the next chairs. But this really happens only very rarely. If you really sleep in the departure area, it is also very safe, because people without a plane ticket won't be able to get in there. You could also sleep outside of the departure area in the public space of the airport. This is generally also very safe, however it could be that you're woken up by drunk party people that wait for an early train since there are some night clubs right next to the airport.
If you want to sleep outside you have several options. You could get one of the airport hotels located in Kloten. There is a bus that will you pick up there. You could also go to the city, especially Oerlikon or the area about the main station are located quite close to the airport. However, all these options aren't really cheap since Zurich is an expensive city.
It's certainly possible to stay overnight as the airport doesn't close. However, reviews online are that it may be hard to sleep. Take this from sleepinginairports:
That, however, was next to one of the Starbucks. Presumably if you go deeper into the airport, you may have better luck in finding a quiet spot:
There is Wifi but it costs, apparently.
While others have mentioned hostels or hotels, there is another option for you at the airport - sleep rooms! They're pricey, but for the novelty could be worth trying ;)