Note the use of the word "accessible" in the original post. This is not a synonym for "pretty" or some way of demanding a particular aesthetic style for its own sake. The reality is that some degree visual impairment is not at all uncommon and that difficulty with contrast can result from any number of conditions.

I'm not sure about the original poster's statement that that site isn't accessible. I haven't tried applying a stylesheet from my browser to it, and I don't use anything to enhance contrast of text in web pages, as my vision is not impaired in that way. There's an argument to be made, though, that modern websites use coloring, layers, and a bunch of other things to achieve layouts, looks and feels to the extent that if you want to see content as it is presented, or to even find it readable or usable at all, you'll have a hard time with applying styles using your browser.

Believing that just because the web can be used differently to black-type-on-white-paper in a wide variety of ways means that every way of using it differently is virtuous is pretty mid-2000s web 2.0 logic. Contrast is useful, and its proven track record for rendering text is no mark of antediluvian shame, of irrelevance or datedness.

Give Opera a try. It has an inbuilt "accessibility layout". Black text on light green background, big font, fixed width.