How to install a gem or update RubyGems if it fails with a permissions error
You don't have write permissions into the /Library/Ruby/Gems/1.8 directory.
means exactly that, you don't have permission to write there.
That is the version of Ruby installed by Apple, for their own use. While it's OK to make minor modifications to that if you know what you're doing, because you are not sure about the permissions problem, I'd say it's not a good idea to continue along that track.
Instead, I'll strongly suggest you look into using either rbenv or RVM to manage a separate Ruby, installed into a sandbox in your home directory, that you can modify/fold/spindle/change without worrying about messing up the system Ruby.
Between the two, I use rbenv, though I used RVM a lot in the past. rbenv takes a more "hands-off" approach to managing your Ruby installation. RVM has a lot of features and is very powerful, but, as a result is more intrusive. In either case, READ the installation documentation for them a couple times before starting to install whichever you pick.
Why do I get ERROR: While executing gem ... (Gem::FilePermissionError)?
after you install RVM you still need few more steps:
Load RVM to the current shell:
Usually this would not be needed if you close and open your terminal again
rvm install ruby
rvm use ruby
The last step is very important as your error message mentioned system ruby not the one controlled by RVM.
You don't have write permissions for the /var/lib/gems/2.3.0 directory
You first need to uninstall the ruby installed by Ubuntu with something like
sudo apt-get remove ruby.
Then reinstall ruby using rbenv and ruby-build according to their docs:
sudo apt-get update
sudo apt-get install git-core curl zlib1g-dev build-essential libssl-dev libreadline-dev libyaml-dev libxml2-dev libxslt1-dev libcurl4-openssl-dev libffi-dev
git clone https://github.com/rbenv/rbenv.git ~/.rbenv
echo 'export PATH="$HOME/.rbenv/bin:$PATH"' >> ~/.bashrc
echo 'eval "$(rbenv init -)"' >> ~/.bashrc
git clone https://github.com/rbenv/ruby-build.git ~/.rbenv/plugins/ruby-build
echo 'export PATH="$HOME/.rbenv/plugins/ruby-build/bin:$PATH"' >> ~/.bashrc
rbenv install 2.3.1
rbenv global 2.3.1
The last step is to install Bundler:
gem install bundler