Medium started to provide custom domain feature in 2015. It was a onetime payment, though it cost $75.
But Medium quietly paused their custom domain feature last December.

The customer support replies: it has been paused indefinitely. We don’t have a timeframe for this at this time. Sorry.

Why does Medium pause it? Because Medium is a curation platform on which all individual writers could publish articles.If top writers and popular publishers could apply custom domain.That means the top writers could take all their traffic away into their own websites in the future. Medium doesn’t want this situation to happen.
In an article, called Medium May No Longer Be a Platform for Publishers or Brands, the author even predicts Medium may pause the Publications feature in the future.

From the point view of writers and publishers, every platform doesn’t exist forever. Some gigantic platforms such as Windows Live Spaces and Myspace even ended their service in the past.
In other words, for someone who desires to construct their brands or publications of interests through Medium should consider other methods.

* setup newsletter service
* redirect your Medium article into your own website
* your profile on another platform