I can understand why some businesses would not take plastic. The fees charge for plastic are high, and profit margins for some businesses are quite low (e.g., 5%). One local market I go to has a minimum for using plastic for that reason.
Seems to be the opposite case here and many other parts of the world. Increasingly, businesses are starting to refuse cash in favour of electronic payment methods. Transaction fees aren't zero, but they are pretty negligible in the grand scheme of things (and the amount is capped by law).
Even sole traders are able to accept card payments because it's cheap and easy, and doesn't come with all the complications of handling cash. You can walk down to a local office supply store, buy a $30 card reader and start accepting card payments in minutes. Card fees are typically under 2%, or capped at some nominal fee like $4 if the transaction is particularly large.
It would be unusual to walk into a shop in Australia where card wasn't accepted. Years ago, a few in my local area tried that and either had their hand forced (through customer domain) or went out of business.
There may be some exceptional edge cases where a business is being transacted simply doesn't have cellular coverage or internet access via some other method, however, I can't say I've ever been to such a place. I've been to some pretty remote towns in Australia where even Telstra (AU's largest telco) coverage is non-existant, and they still have broadband via satellite or the National Broadband Network. It would be highly unusual to go to any town with any kind of population and not have some cellular covarge. Years ago I travelled through a small town called 'Tilpa' in New South Wales. I think it has a population of 40 and the pub is also the petrol station, post office, corner store etc... Even that had 4G/LTE coverage.