Emile de Maat
No, they can simply select new investigators after the Mythos phase (unless the Ancient One has already awoken, and the players are eliminated)(though there are exceptions to that).
Eldritch Horror rulebook, page 11 wrote:
Investigators lose the game if any of the following occur:
*If the Doom token reaches the “0” space of the Doom track, the Ancient One awakens. The back of the Ancient One sheet lists how investigators can lose the game (see “The Ancient One Awakens” on page 16).
*All investigators have been eliminated from the game.
*A card effect causes investigators to lose the game. This most frequently appears on some Rumor Mythos cards.
*The investigators are required to draw a Mythos card but cannot because the Mythos deck is empty.