Is it a good idea to self-fund a postdoc for a year? I am a Ph.D. student in mathematics, and I'm supposed to finish this year. I've been looking for a position next year (not so much in the USA, but mainly in Germany and the UK). While I haven't exhausted all my options, I do feel that maybe it would be a good idea to have a backup plan.

Since I can afford it, I was thinking about taking myself somewhere "productive" and living there off my savings for a year, whilst working with the research group in that university to produce new results. In the meantime, to continue to apply to positions and hope for the best.

My advisor said that this is not a very good idea, but at the same time mentioned one or two people from our field who had done something like that (albeit in the 1970s or 1980s).

How bad of an idea is this? And what sort of strategy I should be considering if I want to move forward with a plan like that?

The alternative, of course, is to go out to the industry for a year or two until I find a position. But leaving and coming back seems like an even worse idea to me.

(I have money for a year in a reasonable place, maybe two if I also work a bit during that time. I also don't have an American, Canadian, British or European citizenship; although my passport is versatile enough that I can move around.)

Clarification. While the question is somewhat general, my field is mathematics. I don't think that not having an affiliation will cause me to be inaccessible to my current network of peers. I have seen several people who did not have an affiliation with any mathematical establishment, but nevertheless continued to collaborate with professors, students and all in all conduct interesting research.

As far as my personal case goes, I also don't think there are others from my field of research who got the positions "and might feel nervous if I'm around". More than implausible, it sounds unlikely.
The job market is extremely competitive, and not having had an official position will not help your CV. This is not the 70s or 80s. Academia is a very different place to what it was then. You should be aware that there is a high chance you won't be able to continue in academia, whatever you do with the next year. Would you be comfortable with running down your savings and then finding you have to change course anyway? If you want to do so, I would strongly advise talking to some tenure-track and newly-tenured professors (or equivalent) about what it is really like. I know some people now are deciding that academia in its current state isn't worth the effort.

For a strategy if you do decide to self-fund:
You need your CV to show the benefit of your extra year. However, applying for maths positions, you don't actually have an extra year, because of the length of the application cycle. To some extent that will help you - you'll roughly still be applying as a new PhD. But you should be looking to have a longer list of good papers. Since getting anything published in that timeframe is basically impossible, I would suggest getting work up on the arXiv, and finding a well-respected researcher in your field as a referee, who feels able to say that they think the paper is of the standard of X journal (or words to that effect). I don't think that's massively easy though.

In addition, I would suggest getting at least one more 'thing' on your CV beyond its current state: organising a conference, writing a grant application, or some solid teaching experience.

The relative merits of going into industry instead will depend on what your field is. For pure mathematics is unlikely you'd be able to move back from industry to academia. In statistics or some applied maths, you can still be seen as a credible researcher on the basis of work in industry. The difficulty is being able to demonstrate your personal contribution to the results. Also, many potential positions will want someone who can teach as well as do research. Demonstrating that would be difficult, but I think not impossible (eg. by supporting colleagues or by tutoring as a part-time job).