Is it possible for Kakashi to obtain an Eternal Mangekyo Sharingan? He doesn't have siblings, nor he is an Uchiha, so the answer would be simple, but as we've seen, he reached a high level in the usage of the Sharingan. so, based on our knowledge of how the Mangekyou Sharingan works, could the combination of his and Obito's eye result an Eternal Mangekyou 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.