Is it possible to Install Android Studio on ChromeBook?

Yes you can.

As of this writing, PixelBook has already been able to run Linux applications including Android Studio. Hopefully this feature will eventually roll out to other high end Chrome OS devices in the near future.

source: Preview Android Studio on Chromebooks

You can, but you are required to install additional linux flavor on your Chromebook i.e, Crouton project. It allows an easy method of doing that.

remember that ChromeOS runs on Linux, and you are basically adding another Linux flavor to your Chromebook.

Some useful links,

• Crouton Github Page
• Crouton Cookbook
• Crouton Google+ Community
• Crouton Script, From an SD Card
• How to Install Ubuntu Linux on Your Chromebook with Crouton
• Howto: Android Development on Chromebook

or you may follow this step by step guide on how to use Crouton to Install an Ubuntu Chroot on Chromebook.

source