The problem here is that the web container needs an oracle client to talk to the oracle database in the db container. This took a bit of doing, since oracle does not provide a client on apt-get, so I downloaded the RPMs from Oracle's site, put them in vendor/ and did the following in the dockerfile:
FROM ruby:2.2.2
RUN apt-get update && apt-get install -y build-essential
RUN apt-get install -y libxml2-dev libxslt1-dev
RUN apt-get install -y libqt4-webkit libqt4-dev xvfb
RUN apt-get install -y nodejs
# Needed for Oracle Client
RUN apt-get install -y libaio1 libaio-dev
# Required for Oracle RPMs
RUN apt-get install -y alien
# Set up app at /code
ENV APP_HOME /code
RUN mkdir $APP_HOME
WORKDIR $APP_HOME
ADD vendor/*.rpm $APP_HOME/vendor/
# Oracle Client Environment Variables
ENV ORACLE_HOME /usr/lib/oracle/12.1/client64
ENV LD_LIBRARY_PATH $ORACLE_HOME/lib/:$LD_LIBRARY_PATH
ENV NLS_LANG American_America.UTF8
ENV PATH $ORACLE_HOME/bin:$PATH
# Set this so you don't have to type it in with rake db:create
ENV ORACLE_SYSTEM_PASSWORD myoraclecontainerspassword
# Install Oracle Client
RUN alien -i vendor/oracle-instantclient.rpm && alien -i vendor/oracle-sdk.rpm && alien -i vendor/oracle-sqlplus.rpm
ADD Gemfile* $APP_HOME/
RUN bundle install
ADD . $APP_HOME/