Today's the first time since 1970 Joe Biden is not holding a public office
But technically
I don't want to nit-pick, but according to Wikipedia
Biden served on New Castle County Council 1970-1972.
Elected U.S. Senate 1972, was sworn into U.S. Senate Jan 3 1973, so there was a gap in between New Castle County Council and U.S. Senate where he held no public office.
Elected Vice President 2008, resigned from U.S. Senate Jan 15 2009, sworn in as Vice President Jan 20, 2009, so there was a 5 day gap where he did not hold any public office.
But like I say, I do not wish to nit-pick.