I am trying to produce a page that filters something by multiple taxonomies at once—so, for instance:

http://blog.com/category/news/tag/politics

to obtain a page of news posts about politics.

Is even possible? (I am not wedded to the particular URL structure)

Version 3.1 of Wordpress introduced advanced taxonomy queries which turn to just do what you are after.

Find out more about them here:

http://ottopress.com/2010/wordpress-3-1-advanced-taxonomy-queries/

If you would like it to have an advanced search form. I authored a tutorial, but is within the spanish language.