Since the Big Bang did not begin at one place, did the universe
begin at
one time, but everything was just a little closer together ?
Yes, that's correct. There's no special place in the universe, since the Big Bang happened everywhere, but there is a special time. Recent measurements of the Cosmic Microwave Background (the leftover glow from the Big Bang) by the WMAP telescope puts the birth of the universe at 13.7 billion years ago.
Answers provided by HSU Astronomy Professor David Kornreich.