Is it self-plagiarism to reuse my own unpublished material? If I reuse unpublished work in new unpublished work, would this be considered self-plagiarism?

Examples would be:


Use the same paragraph of text in more than one cover letter for job applications
Use the same unpublished results in more than one funding application


I would usually say better safe than sorry but it just seems silly to state I previously used this text/material in [unpublished work X], in contexts such as the examples I gave above.
No. Unpublished work is unpublished.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If your unpublished work has been circulated among other authors and has been cited by them, maybe you can cite it too.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.