This is only partially true.
Web Designers are under the obligation to make their website as accessible as possible, i.e. without discrimintion. However , this does not mean making a site truly accessible for everyone, this would be far too time consuming.
Blind people will make use of Screen Readers, and so therefore the W3C have said that to make websites accessible for screen readers, content should be placed in a sensible order with the use of DIVS and NOT tables.
Tables that are incorrectly used (for layout) pose problems, for screen readers.
Making sure that your font can be changed is another large accessiblity issue, aswell as proper use of "TITLE" and "ALT" tags to give extra info about images (if images are disabled) for readility reasons.
AccessKeys can also be used to aide with navigation on a site.

General websites should be accessible and in theory, may be in violation of the Disability Discriminations Act, but it is only large public websites that need worry with regards to being prosecuted. It is the more common websites that come under scrutiny, so even though you should design for the largest amount of people possible, you can get away with not doing.

Craig