According to ComicVine, Barry Allen named himself after Jay Garrick's Flash. Since they existed in the same universe, what happened to Jay Garrick?
That Jay Garrick was a comic character of Earth-2... not a real person on Barry Allen's Earth-1.
In 1956, DC Comics reinvented the Flash character, giving him a new costume, name, and background. Named Barry Allen, he was completely unrelated to Jay Garrick. In fact, Garrick never existed as far as the new books were concerned. Barry Allen's first appearance shows him reading a copy of Flash Comics, lamenting Garrick was "just a character some writer dreamed up".
This is actually stated at the same ComicVine link:
He [Barry Allen] also had a love of comic books, his favorite hero being the Flash ( Jay Garrick).
In 1985, DC Comics merged all of its fictional characters into a single shared universe. Jay Garrick now shared the same world as the new Flash.
So, in answer to your question, they didn't actually exist in the same universe/Earth variant until much after Barry Allen's Flash was created.
You can read about the current "status" of Jay at the Wikepdia link.