I recently played a game where I built nothing but tiny and small ships the entire game, and they never left the orbit of my planets.
It helps if you have a good diplomacy skill. I was playing as the Terrans against the Alterians, Torians, Yor, and Drengin. And I basically managed to become friends with the Drengin and Torians.
My strategy basically consisted of trading (non-military) techs with the guys I was friends with, and using the money/techs I got from them to bribe people. The Alterians hated me and were on one side of me, and the Yor, who also hated me, were on my other side.
As people have said, establishing trade routes helps alot. But even if people do like you, they will declare declare war on you despite your trade routes.
So, I focused on developing Diplomacy techs (for the bonuses, and government changes to keep my economy doing really well), and Research techs (to stay ahead in the tech race).
Basically the entire game I made sure I kept an eye on my relations. Any time a civ started sliding toward "cool", I would bribe someone to attack them. And it's usually best to pit good civs versus evil civs and vice versa, since they will be more likely to attack each other. I bribed the Alterians to attack the Yor, the Drengin to make peace with the Torians and attack the Alterians, and when i couldn't convince the Alterians to attack the Yor, I paid off the Torians to do so. At one point I even convinced the Yor to attack the Drengin.
The whole time, I kept building up my culture, spamming cultural starbases. after a while, i finally had everyone attack the Yor to wipe them out... and they surrendered to me, since--even though they didn't like me--I was the only race that hadn't attacked them, and had traded with them the whole game.
My influence ballooned and I ended up with an influence victory, without fighting a single war.
It was actually one of the most fun games I've had. It was immensely satisfying playing the other civs off each other like that.