Over the course of time I got annoyed by how long it takes to properly install non-latin language support for a browser, so here’s a (hopefully quick) way to get it working.
Note: For the changes to take effect, I had to restart my browser. I’m using Chrome 23 on Debian with XFCE.
According to Wikipedia’s “Multilingual support (East Asian)”, you need to install these packages for Simplified and Traditional Chinese, Japanese, Korean and Vietnamese.

sudo apt-get install ttf-arphic-uming ttf-wqy-zenhei ttf-sazanami-mincho ttf-sazanami-gothic ttf-unfonts-core

That worked perfectly for me, let’s move on to Multilingual support (Indic). On my installation I was missing support for Kannada, Telugu, Tibetan, Khmer and Burmese.

sudo apt-get install ttf-kannada-fonts ttf-telugu-fonts fonts-tibetan-machine fonts-khmeros fonts-sil-padauk

You can also install a package called fonts-indic, which supposedly includes support for all Indic languages. I did not try that and don’t know if it really includes everything.
In the “Multilingual support” article you can also find a language or typeset called Deseret, which I didn’t find a package for. I will leave it uninstalled for now.