Its possible to survive with using email and online banking, but it isn’t easy. If you avoid using credit or debit cards, you won’t need to check balances online to prevent identity theft (a.k.a. crooks raiding your account). The law still measures the period to report fraudulent transactions based on printed statements, though the card will usually stop working long before that. Some banks still send printed statements and return cancelled checks, but they might charge extra. Non-digital media do report news, weather, etc., albeit more slowly.
Obviously some lines of work are impossible, such as anything to do with finance, law, sciences, most retailing, etc. Applying for jobs without the internet is also a problem as most large employers expect applications to be online.
It might be easier to have one computer in an open area (no “sneaking a peak” without other family members knowing it) and limiting what it can be used for.