This is true for settlement, but not authorization. When you spend your money it could be hours or days until the transaction is correctly reflected in your balance. But in order to spend it in the first place the transaction has to be authorized, and this happens in real time. Depending on where you live, who you bank with, what kind of card you’re using, how you’re using it... this authorization may rely heavily on your bank being available for online transaction processing, or not very much. But unless you’re doing a transaction with one of those imprint devices, your transaction must be authorized in real time by somebody.
I carry some cash too. But the times I’ve needed it were for things like an outage of the internet connection at a restaurant, or an issue with a payment terminal provider (or for the very uncommon cash-only merchant).
What makes banking systems reliable is that there's so much built in latency in every operation that nobody really notices if something is down for six hours.