What is a Self?

It is often claimed that the self does not exist, that there is no such thing as a self, anywhere.

But what can possibly be meant by such a claim?

Suppose that I were to say, "There is no such thing as a sklorb."

And someone asked me, "What is a sklorb?"

And I replied, "That's just it: sklorbs don't exist!"

And they said, "Yes, but what do you mean by 'sklorb?'" 

And I responded, "The very concept of sklorb is meaningless."

Them: "Then what are you talking about, when you claim that sklorb doesn't exist?"

Me: "Nothing."

Them: "Then why did you say it?"

Me: "No reason."

The word "self" must mean something, in order for the sentence "There is no self" to be meaningful at all, in just the same way that the word "egg" must be meaningful, for the grocer to meaningfully inform me that they're all out of eggs. 

How about this dialogue?

Bill: Joe Shmoe's self has no substantive content.

Jill: Compared to what?

Bill: What do you mean?

Jill: Compared to whom? 

Bill: uhh...?

Jill: Are you saying that someone else's self does have substantive content?

Bill: No.  No self anywhere has any substantive content.  The self is an illusion.  

Jill: Compared to what?

Bill: Nothing.  All selves are illusions.

Jill: Compared to what?

Bill: I don't see what you're getting at.

Jill: Maybe when you say that "The self doesn't exist," or "The self is an illusion," all you mean is that your idea of the self doesn't exist.  Your idea of the self is an illusion.  If one self were a "fake," so to speak, that would be meaningful, just it could be meaningful to say that a particular dime is a counterfeit dime.  But if you're claiming that all dimes are counterfeit, to me, it sounds like you're confused about the definition of the word "dime".  It sounds like you have some concept of a "true dime" in your head, which doesn't exist, and you're measuring all existing dimes against that ideal, and they all fall short - but perhaps that's a simple mistake on your part.  Maybe these things, which you regard as "fake" dimes, are exactly what dimes are, and nothing else.  The concept you have of a "real" self, with some kind of substantive content that you feel is missing, is an abstract fantasy - but these things, which human beings all around you have - those are selves.  That's what "self" actually means.  These funny, paradoxical, transitory things, which are nothing but constructs - those are indeed real selves.