I've yet to have this happen, but I've had situations where I just *barely* got one resource (horses) hooked up. If you find yourself in a situation with none of the 3, I think you have 2 options: research archery in a hurry *or* restart the game. You're not going to take out another AI with archers, but the goal here is survival. Build enough archers to make the AI think twice about attacking you. If I survived until construction, I would build catapults, more catapults, and even more catapults (unless I had ivory, in which case I'd build war elephants too.) I would be running vassalage/theocracy so that each cat got 2 promotions from the time they were built. Some would get accuracy (for knocking down city defenses in a hurry) some would get barrage II (for max. collateral damage) and some would get city raider II. Once I had enough catapults, I'd send them off, accompanied by a few archers with useful promotions to help defend the cats against horse archers. Then I'd go out and capture a nearby city with the resource I needed.
As for the barbs, I've found that they do seem to vary from game to game. I've had games where I only saw 1 barb warrior for the whole game, and I've had games where I had axes/swords/archers popping all over the place. The best advice I can give is to send out fog-busters. Station troops on a hill (preferably a forest/jungle hill) outside your borders. You want to be far enough away to see as much terrain as possible (to lower the frequency of barb appearances) but close enough there's no fog between the borders and your fog busters. Positioned like this, warriors can hold off barb archers and archers can hold off barb axes/swords/spears. It's been my experience that, even when the odds are ridiculously in my favor (barb warrior vs. woodland II axe on a forest/hill) they still head for my fogbusters rather than bypass them.