Is it possible for Kakashi to obtain an Eternal Mangekyo Sharingan?
The answer to your question is yes, Kakashi can obtain Eternal Mangekyou Sharingan.

But the problem is that his Sharingan consumes a lot of his chakra. At some stages, his overuse of Kamui made him exhausted, too. In this source, it does say that:


  "Repeated use of the technique puts a great deal of strain on the
  user, causing his left eye to bleed if overused"


As a result, using Eternal Mangekyou Sharingan could result in a life-threatening situation for him. And as per the previous answer, you need to transplant a Mangekyou Sharingan, preferably from somebody close, such as a sibling.