Google is currently accepting applications for a new beta that will allow testers to mark the non-Google files they want for offline access. Unfortunately, only G Suite organizations can apply to be part of the testing period, and those who do get the chance to use it early can only take advantage of the feature if they're using the Chrome browser. They have to enable offline mode from within the Drive or Docs settings first and then sign into the account associated with the approved G Suite organization before they see the option to save files offline.
Google says the current version of the feature doesn't work with Chrome OS, but it's working to bring it to the platform in the future. No word yet on whether it's also coming to other browsers other than Chrome -- let's just keep our fingers crossed for wider compatibility when it's ready for rollout.