<The Democrats could put up a dead man for_ local office_, and he'd win hands down and, come to think of it, that's precisely what they've done in all too many cases. >
1. What does local office mean here?
2. Why no article before it?
Could you please enlighten me?
1. It means "a local political position".
2. In the phrases "run for office" and "put up for office", it is not necessary to use an article before "office". This gives a sense of running for any position in general. If you include an article ("run for an office"), it then gives a sense of running for a particular office rather than any office in general. In this case, they weren't taking about any one office, but rather all offices in general, so no article was used.
It's called being true to your name and just accepting that no articles are used before such a word and trying to pin down the meaning of that word and imitating the native speakers' usage of that word.
A local office would be an office at the city level: mayor and below.
Some addition for the office in question:
<I see that Cindy Sheehan is announcing a possible run against Speaker of the House Nancy Pelosi.>
beneficii , I know very well English abounds in inconsistencies (or at least what seems to us like those) which foreign learners just have to accept as real usage, but I always try to find out some logic.