Getting down to the real nitty gritties
Having done your best to avoid the barriers, there are some things you can do to actually improve the accessibility of your site generally and also the accessibility for people with disabilities. Some countries now make website accessibility mandatory for governmental public information sites and for others, it is just makes good sense.
If you are just providing information, it should be quickly and easily found, understandable, be readable on any device that can access the Web and as devoid of extraneous matter as you can possibly manage. Don't hide the site, its functionality or its information.
Then, consider how people who are visually impaired can use the site. There are ways to convert a Web page into speech but they require software to interpret the source code behind a Web page and don't actually read the screen as we do.
Once it has stripped-out all the HTML tags, the software reads out the text content of the page along with the textual descriptions of images provided in 'Alt' tags. If you have images on a page, they should have Alt tags - even it is only 'XYZ Logo'. If you don't want the Alt tag to be read out because the elements are just page furniture, put Alt="" - just the quote marks with nothing inside.
A page constructed using CSS instead of tables will generally have a more logical flow of text in the source code. Tables don't present text in the source code in the same way they do on the screen. Have a look at the source code of a table and you will see what I mean. Having said that, screen reading software can cope with simple tables and older browsers do not handle CSS well.
If the style of your page is important, and your audience is likely to be using older browsers, (at the minute, a lot of people still do), relying totally on CSS positioning can be dangerous and render the page unreadable to many, totally outstripping any advantages.
Of course, not all visually impaired people need screen readers. Some are quite happy with text if they can make it large enough on their screens. If accessibility is important to your site, you should use relative rather than absolute type sizes. In this respect, my pages here with their fixed type size are bad for general accessibility, but then it is not aimed at a general audience.
People with poor eyesight or colour blindness also prefer their text to have good tonal contrast. Just because you can easily read that red text on a green background perfectly well, the ability to distinguish red from green of the same tonal value is not so easy for those with color blindness problems.
If you can, switch your monitor to greyscale mode so that you are looking at tonal and not colour contrast. Some computers don't support greyscale, in which case, viewing your colour screen through a piece of strong blue glass, plastic or acetate will give a similar effect.
The other thing to consider is that not all disabilities are about vision. People with learning disabilities and dyslexia, for instance, can benefit greatly from 'visual aids'. Using color-coding to help separate and identify elements together with other visual clues such as icons and images go a long way to making 'plain text' more approachable and understandable.