According to the page linked below. In order to play minecraft you wpuld have to root your device to the linux kernel, and even then it might not work. Chromebooks run off the chrome web browser, so if your game requires an OS, it probably won't run off this computer.
whoa yes you can i did by installin full-blown linux on my chromebook. I used this method if you feel youre up to "hacking" your chromebook a little. heres the link you like to try installing linux , & you don't really have to ROOT the chromebook because the acer one is "hacker friendly" just boot into developer mode & it will let you install linux with no trouble. http://chromeos-cr48.blogspot.com/2013/05/chrubuntu-one-script-to-rule-them-all_31.html
No and yes. Minecraft doesn't run on ChromeOS natively. You can use a flash version, use a remote log in to a Windows/Mac/Linux system that can play Minecraft, or you can install something like ChrUbuntu or Crouton which leverages Chrome OS's native Linux kernel and installs a full Linux UI on top.
http://www.howtogeek.com/162120/how-to-install-ubuntu-linux-on-your-chromebook-with-crouton/ is a good walkthrough if you want to install Crouton. Crouton will be faster, but installs in local storage so it's best for the 320gb model. ChrUbuntu can be installed on either the local storage or a thunbdrive/memory card, but it's slower and requires dual booting while Crouton runs side by side (it's basically another UI installed on top of Chrome OS).