<p>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?</p>

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
!