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:

cd $HOME
sudo apt-get update 
sudo apt-get install git-core curl zlib1g-dev build-essential libssl-dev libreadline-dev libyaml-dev libsqlite3-dev sqlite3 libxml2-dev libxslt1-dev libcurl4-openssl-dev python-software-properties libffi-dev

git clone https://github.com/rbenv/rbenv.git ~/.rbenv
echo 'export PATH="$HOME/.rbenv/bin:$PATH"' >> ~/.bashrc
echo 'eval "$(rbenv init -)"' >> ~/.bashrc
exec $SHELL

git clone https://github.com/rbenv/ruby-build.git ~/.rbenv/plugins/ruby-build
echo 'export PATH="$HOME/.rbenv/plugins/ruby-build/bin:$PATH"' >> ~/.bashrc
exec $SHELL

rbenv install 2.3.1
rbenv global 2.3.1
ruby -v

The last step is to install Bundler:

gem install bundler
rbenv rehash

Then enjoy!

Derek

If you want to use the distribution Ruby instead of rb-env/rvm, you can set up a GEM_HOME for your current user. Start by creating a directory to store the Ruby gems for your user:

$ mkdir ~/.ruby

Then update your shell to use that directory for GEM_HOME and to update your PATH variable to include the Ruby gem bin directory.

$ echo 'export GEM_HOME=~/.ruby/' >> ~/.bashrc
$ echo 'export PATH="$PATH:~/.ruby/bin"' >> ~/.bashrc
$ source ~/.bashrc

(That last line will reload the environment variables in your current shell.)

Now you should be able to install Ruby gems under your user using the gem command. I was able to get this working with Ruby 2.5.1 under Ubuntu 18.04. If you are using a shell that is not Bash, then you will need to edit the startup script for that shell instead of bashrc.

(January 2019) To install Ruby using the Rbenv script, follow these steps:

1. First, update the packages index and install the packages required for the ruby-build tool to build Ruby from source:

sudo apt-get remove ruby
sudo apt update
sudo apt install git curl libssl-dev libreadline-dev zlib1g-dev autoconf bison build-essential libyaml-dev libreadline-dev libncurses5-dev libffi-dev libgdbm-dev

2. Next, run the following curl command to install both rbenv and ruby-build:

curl -sL https://github.com/rbenv/rbenv-installer/raw/master/bin/rbenv-installer | bash -

3. Add $HOME/.rbenv/bin to the system PATH.

If you are using Bash, run:

echo 'export PATH="$HOME/.rbenv/bin:$PATH"' >> ~/.bashrc
echo 'eval "$(rbenv init -)"' >> ~/.bashrc
source ~/.bashrc

If you are using Zsh run:

echo 'export PATH="$HOME/.rbenv/bin:$PATH"' >> ~/.zshrc
echo 'eval "$(rbenv init -)"' >> ~/.zshrc
source ~/.zshrc

4. Install the latest stable version of Ruby and set it as a default version with:

rbenv install 2.5.1
rbenv global 2.5.1

To list all available Ruby versions you can use: rbenv install -l

5. Verify that Ruby was properly installed by printing out the version number:

ruby -v

# Output
ruby 2.5.1p57 (2018-03-29 revision 63029) [x86_64-linux]

SOURCE: How To Install Ruby on Ubuntu 18.04

EDIT: Install rubygems:

sudo apt-get install rubygems