With just these three files, you can run a sinatra app with redis using Docker.

app.rb

require 'sinatra'
require 'redis'

set :bind, '0.0.0.0'

Cache = Redis.new(host: 'redis', port: 6379)

def get_hit_count
  Cache.incr('hits')
end

get '/' do
  "Hello World! I have been hit #{get_hit_count} times.\n"
end

docker-compose.yml

version: '3'
services:
  web:
    build: .
    ports: 
      - "4567:4567"
    volumes:
      - .:/code
  redis:
    image: "redis:alpine"

Dockerfile

FROM ruby:2.5.1-alpine
WORKDIR /code
RUN gem install sinatra
RUN gem install redis
CMD ["ruby", "app.rb"]