Answer:
You need about 1GB of RAM to run Chrome along with any other applications open. Right now I have 9 tabs open and Chrome is the only application I am running right now and I'm using 650MB of RAM if that gives you some perspective. I don't have a lot of applications running in the background either.
There are some tweaks to minimize the amount of RAM Chrome uses, but I would personally install a SSD hard drive. Then you won't have to worry about the amount of RAM so much because the swap file is 10x faster with a SSD. I only have 512MB of RAM but I have a SSD and my old computer flies!