I struggled with this for awhile, abandoning it midway through, but I finally got it to work. Most of the instructions for installing typo 4.0.1 on dreamhost by Aiden Bordner worked for me, except you need to edit db/migrate/051_fix_canonical_server_url.rb as described by Chris H.

To summarize:

1. Download Typo using Subversion:
   svn checkout svn://typosphere.org/typo/trunk typo
2. Setup a new host and set the directory to /home/username/typo/public and enable FastCGI support.
3. Setup a mySQL server.
4. Create config/database.yml:
   login: &login
  adapter: mysql
  host: mySQL server hostname
  username: mySQL server username
  password: mySQL server password

development:
  database: typo_db
  <<: *login

test:
  database: typo_db
  <<: *login

production:
  database: typo_db
  <<: *login

5. Edit config/environment.rb and uncomment the line that says ENV[‘RAILS_ENV’] = ‘production’
6. Edit db/migrate/051_fix_canonical_server_url.rb and comment out or delete the line b.settings['canonical_server_url'] = b.settings['canonical_server_url'].gsub(%r{/$},'')
7. Setup your database by running rake migrate RAILS_ENV=production. Note that this step will fail if you don’t edit db/migrate/051_fix_canonical_server_url.rb
8. Edit public/dispatch.fcgi and insert the following code before the line that reads RailsFCGIHandler.process!
class RailsFCGIHandler
  private
    def frao_handler(signal)
      dispatcher_log :info, "asked to terminate immediately"
      dispatcher_log :info, "frao handler working its magic!"
      restart_handler(signal)
    end
    alias_method :exit_now_handler, :frao_handler
end

9. Make sure that both public/dispatch.rb and public/dispatch.fcgi are executable.

Typo can sometimes be excruciatingly slow to load up, you may have to reload the page a couple of times.