In my case, I solved doing just what the error message suggests:
Your user account isn't allowed to install to the system RubyGems.
You can cancel this installation and run:
bundle install --path vendor/bundle
to install the gems into ./vendor/bundle/
So, instead of:
bundle install
I ran:
bundle install --path vendor/bundle
That was the solution for this guy.
The downside of that solution is that it creates a vendor folder inside the current folder, which can be added to .gitignore if it is to distribute the application through Git.