I just played a whole game as the Aztecs (on Earth... builder/space ship type game)
I never built a fighting unit... only diplomats, settlers, and caravans. Barbarians are a steal, literally... you can buy them for a song, knights for ~50 gold... rush completing a barracks/switching to a knight yourself would cost just under 80 gold (assuming you had a shield or two to start). And cheap sails long before you'd have the tech yourself... unless you went right for it (which I wouldn't as I was playing isolationist... even my trade routes were local). There is the problem of not being able to buy units that are stacked, but if you have a couple already bought and maybe a couple units from huts, you should be able to take out the stack. Pop huts with a pack of four diplomats and buy yourself a cheap army if you get barbs. (I used to live in fear of the 8-barb hut explosion... now it's like "hey, instant army!")
The great thing about diplomats (and caravans) is that they don't have maintenance costs... so even though you have to have a little cash on hand to buy off attackers, having a diplomat instead of a phalanx defending your city is like +1 shield per turn forever.
Buying is always successful, so my favorite part about this strategy is that there isn't that annoying luck factor involved. Fighting is gambling... you could lose your 40 shield plus 1 per turn chariot even when the odds seemed in your favor... but your 30 shield plus 0 per turn diplomat will always be able to "win" the battle if you've got enough cash.
When the English came around, I bought their stupid triremes, for a pittance. When the Babylonians finally showed up off shore, I bought their ironclads. Theirs were a little expensive, but then I had my own ironclads at my disposal... when you buy a unit, it's a net 2 unit swing in your favor... you had 0 ironclads, they had 1, now they have 0 and you have 1. So even if it's twice as expensive as buying one in your own city, it's still break-even... and you can do it even regardless of peace treaties. And it wasn't quite twice as expensive, though it may have been close. Not that it mattered... the ironclads didn't show up until well into the first millenium AD, and I had so much cash by then a hundred gold wouldn't have made much difference.
I ended up having to wait until 1490 to launch my spaceship, because I built it in the 1200s before I found out (searching threads here) that I couldn't be successful with a full ship until 1490. In 1500, a turn before the end, a cruiser showed up a couple squares offshore. The game ended before I could buy it though.
Oh, and diplomats are great for early exploration... when you switch to democracy you don't have to hurry home or disband!