This is the near to listener that, right? So how could we write the far from both that in this sentence?저의? But that would be my, right?
Nope. 거의 means 'almost'. There are no dogs would be 개가 없어요. But there are almost no dogs would be 개가 거의 없어요.
Why not "there is almost no dog"?
That's not grammatical English. When talking singular in English it's either "there is a dog" or "there is no dog". There is no between "almost".