Google is now offering repair service for its products for those that are unable to make it to a locally supported third-party repair center. The process isn't the most expedited and could take more than two weeks.
The way to get things started is fairly simple, you head to the repair website and enter in the IMEI number for your phone. Google is currently only supporting the repair of the Pixel, Pixel XL, Pixel 2, and Pixel 2 XL. Once the model is verified, Google will give you a detailed list of problems to choose from so you can properly address the issue or issues you are having with your phone.
The next menu will allow you to choose the shipping method, either using your own packing material or waiting for Google to ship out its own box. At that point, you will need to agree to some terms before being able to ship out your unit. Although Google is the front, it looks like the service is provided by KMT Wireless and Cynergy, as can be found in the fine print. This service is currently only available to those in the United States.