How did society function before banking?

I think most people kept whatever money they had hidden somewhere and only took it out when it was needed. Otherwise, they used the barter system and no monies were exchanged.

I assumed under their matteress......

Barter maybe.  I raise cattle .. I'll give you a cow if you build me a cellar.

I'm guessing that on Monday they invented currency, and on Tuesday someone said "invest it wixth me". 

Banking in some form has been around as long as currency.

And currency in some form has been around as long as civilization.
Actually, you would have to have a form of currency prior to building the first city.