There is never no hope in a position. If you are good at negotiating, you can usually talk your way out of elimination - even if this doesn't work, it's worth trying.
The obvious example of why never to give up is the czech's comeback from 1SC to a solo:
http://webdiplomacy.net/board.php?
gameID=34872I have a couple of games on vDiplomacy in which I got into a draw despite being extremely close to elimination (you have to copy the whole url, not just click the link):
http://www.vdiplomacy.com/board.php?
gameID=9722http://www.vdiplomacy.com/board.php?
gameID=7006And although survives are not a great outcome, given that I could not affect this game's result whatsoever I think it was a good outcome for me:
http://www.vdiplomacy.com/board.php?
gameID=6064#gamePanel
You should always play on even if you think a situation is hopeless - it very often isn't.