That is not true, though? Or rather, banks can indeed shut down, but that's what the Federal Deposit Insurance Corporation is for. The FDIC insures up to $250,000 per customer, per bank. So there is not much safer place to keep emergency funds than a US bank.
actually banks are one of the worst ways to store money for real emergencies. In the event of a bad recession you will quickly find the banks are more fragile than you thought.
Multiple times in history people who thought the banks couldn't fail saw they crashed hard.
Furthermore as other posters said, inflation can make you money worthless quickly.
If you want an emergency fund, and I mean societal collapse proof funds, get physical wealth and store it.
Silver and gold have been useful in times of war, people who lived through the bosnian genocide said they used physical wealth that was still useful to get necessities.
I am not trying to be a prepper nut, but its good to prepare somewhat for a bad recession or a problem with violence that could happen.
just get a little, a few bars, and you are set incase anything goes wrong.
It'll never go bad in value, it'll never be irretrievable due to financial problems, it'll never be destroyed by rot.
Take physical wealth and move somewhere else if a really bad one happens.