Is it self-plagiarism to reuse my own unpublished material?

No. Unpublished work is unpublished. It's almost as if it doesn't exist. Given that it is your work, there is no case of plagiarism. I don't even think you need to cite yourself --- what would you actually be citing?

Indeed, how would you prove that you plagiarised yourself?

If your unpublished work has been circulated among other authors and has been cited by them, maybe you can cite it too. But that seems a bit odd.

Self-plagiarism is attempting to pass off your own words as previously unpublished original work when they are not. If you are copying text from a work that is not intended for publication, then it cannot be a prior publication, and thus it cannot be self-plagiarism. Likewise, it is OK to take text from one of your rejected works if you have decided not to publish it. If the other work is not published but is also aiming for publication, however, then it would be self-plagiarism (though which of the two works would be "original" and which "self-plagiarized" could depend on the details of the progress of each toward publication).

Many pieces of text, however, do not expect the work to be original, in which case you can plagiarize, but not self-plagiarize. Taking your example: if you create job application letters by modifying a base template, the duplication from letter to letter is perfectly ethical, because there is no expectation of originality in a job application, just for truthfulness.

Funding applications are a funny in-between state. They do not have the same expectation of originality in text as a publication, but most funding agencies require that you not be attempting to fund the same proposed research through any other application. Thus, you would be safe to use the same preliminary work text as part of pitching two different ideas (though it will probably need to be customized for context in any case). Pitching closely related ideas, however, even with different text, would typically be a violation of the funding agency rules. Since there is a decent chance that some of the same people will see both proposals, it's a fast track to rejection and burning bridges with funders. Don't do it.

Yes, technically you CAN plagiarize yourself.

According to this page on the Lancaster University's page, for example, they say: "Duplication of your own work:It is possible to plagiarise yourself by directly copying a section from a previous essay you have written and submitting it as part of your new essay."

You need to check the student policies of the school you attend, most likely there is a clause that specifically forbids you from handing in two separate essays that share sections containing the same work.

I'm not sure if this is really what you are asking though, because you make it sound more like you will be paraphrasing previous work within a new context, which I believe would be acceptable use.