Is it possible to install linux on a steam machine?

The Steam Machines all run a custom version of Debian called SteamOS.

While usually you are jailed within the Big Picture mode, Valve does provide a method to escape to a full-on Linux desktop:

1. Head over to the Settings menu:

2. Select Interface and check Enable access to the Linux desktop:
   _{(source: howtogeek.com)}

3. Now, you can use the Exit command to quit to the desktop:
   _{(source: howtogeek.com)}

From here you have access to all of available Linux, including a terminal and whatever apps you want to install. Do note, however, that you have left the safe-haven set up for you by Valve, and you are responsible for your own actions out here.

^{Images and steps sourced from HowToGeek}