What things have to be done before I can honestly tell myself my web application is accessible by anyone? Or even better, convince Joe Clark. I don't have any video or audio to worry about, so I know I won't need transcripts. What else do I have to check?
Your question is very vague, but in short, you need to ensure that your site meets one of the three levels (A, AA, or AAA) of the Web Content Accessibility Guidelines.
FWIW, in my experience, if you are providing anything other than a purely static HTML site, aim for AA. Trying to follow the WCAG guidelines stringently to triple-A standard for a dynamic website is the road to hair loss IMO. This may change with WCAG 2.0.
Good luck!
EDIT: @Blowdart suggests running your site through online checkers. This is fine so long as you realise that many of the WCAG guidelines (especially towards the higher end) are so arbitrary, they can only be validated with a human eye.
Do not trust the output of these online checkers and blindly stick a AAA badge on your site. If you are called on it, you may be in trouble.
+1 Blowdart for suggesting HTML and CSS validation, and Chris Pederick's add-on is great!
Run it through a bunch of checkers. Check the HTML is valid. Try to use it with script disabled. Try to use it with CSS disabled. You can do all of this with the web developer addon for Firefox
If the site is usable with JavaScript and CSS disabled, then you're doing pretty well. The Web Developer toolbar in Firefox lets you easily disable both of these. Another way to check is to use Lynx, a command-line, text-only browser. Beyond that, your best bet is to check the site heuristically against whatever guidelines apply (in the U.S., that's usually Section 508). No site will be perfectly accessible, but doing these things will ensure that yours is very accessible.
You should also check out the WAI-ARIA stuff:
- http://www.w3.org/WAI/intro/aria
- http://alistapart.com/articles/waiaria
- http://juicystudio.com/article/wai-aria-live-regions.php
And, for a perspective on the challenges actually faced by users with various disabilities when they try to use the web, have a look at some of the presentations and videos from the Scripting Enabled conference in London on Friday: http://scriptingenabled.org/ (I don't think all of them are uploaded yet)
Online accessibility checkers can help, following the WCAG can help, trying your site with demo versions of screen readers can help, but the only way to "insure" a site is accessible is to have an expert check it for you, or become an expert your self. Fortunately, if you do follow the WCAG 2.0, test with a screen reader, and run the online tests, 98% of the time, you will be just fine. But if you want a guaranty...