You could apply to a post-doc without a high-school degree -- an application only involves sending in the application package. No one is going to stop you from mailing it in.
That says nothing about whether or not you will get the post-doc, let alone if the secretary will just toss it in the recycling bin before it even gets seen by the faculty.
The real questions are:
- What are the eligibility requirements of a post-doc?
- How competitive is the post-doc?
For the first question, most post-docs require a PhD in hand or late-ABD status. Most do not require publications and if they do, they will state that clearly on the application page.
The second question is the real question. Post-docs are usually extremely competitive. In order to be a finalist, you have to be n+1 of whatever quality and quantity the faculty is looking for.
If your peer group has no publications (0) and you have 1 publication, you will stand out as the +1. Since it's not possible to have -1 publications, it's hard to stand out from a group with 0 publications to your name. If your peers have 10 publications, you need to have either more in number or more in quality, or something else that your competitors don't have.
For example, some post-docs in some fields value other things than research publications. I occasionally read for post-docs in the humanities and there the quality of your dissertation and proposal is more important as most post-docs in those fields apparently have zero or near zero publications at the ABD stage.
And finally, there are some post-docs that just aren't very competitive -- either because of geography or topic or paucity of funding or other reasons.
I would also be remiss if I didn't mention the social/human factor. Many people get post-docs on the strength of their recommendation letters, as Pete Clark mentions. Strong recommendations attesting to your brilliance and to the problems you faced with your current project would go far in convincing a postdoc committee to take a chance with you. You could also try to ask your supervisor to use some of her/his social capital to try to persuade a colleague at another university to take you on.
Your mileage may vary.