Well I never came out with this idea publicly, mainly because I haven't tested it until now. I think the way we can get back in the game, is by taking a big risk. We research the most advanced tech we can (as long as the other civs don't have it.) Once we finally research it (which we will have to devote a lot of money to it) we open ourselves up to trade.
When we aquire said tech, we go to each civ, and ask to trade said tech for a tech we don't have. If we know 4 civs, it's possible to get 6 or so techs out of them.
I tested this in a game on the diffuculty of prince (wait maybe it was higher) however. I was in the same situation (although out of all the civs I knew (7) I was in last.) The first advanced tech I researched, I traded away to almost every civ, and in one turn I gained 6 techs. The turn after that I gained more. This put me back in the game with a middle score. I then researched another tech, since I was now more caught up, and this time got 3 techs, and a lump sum of about 450 Gold.
Not only could we be extorting techs, but we would be gaining superior miltary technology. I think we are far enough behind to start thinking a little more radical (which is ironic, since I can't stand liberals)