This means Ruby cannot find Xapian. Be certain you have installed Xapian with the Ruby bindings for the current version of Ruby you're using.
I get UnimplementedError: This backend doesn't implement spelling correction
This means Xapit is using an in-memory Xapian database and spelling is enabled. This usually happens when the configuration file wasn't loaded properly. Make sure the config/xapit.yml file has been generated (with rails g xapit:install) and that it contains the correct database_path and spelling options.