Well you're not doomed, but you're not in a great spot either since it's already 700 AD. Diplo victory seems easiest here (well it's always the easiest probably...)
Unless you want to try culture, fill out your peninsula. Also, I'm very surprised you have not explored at all, there have got to be islands out there with good spots to build cities.
If you aggressively expand, set up more cities, hopefully have some maritimes, you should be able to kick your economy up a notch. Spam research agreements, + increased population for more tech and you should be able to stay at least close to Alexander. This is all you need, a citadel chokepoint is an incredibly effective defense. Just make sure to control the seas (easy with current AI).
Anyways the computer is fairly bad at winning, so unless you get unlucky you can probably explore the world and win a diplomatic victory before they even get close to culture or space.