The sites archived have to meet certain criteria but you can suggest a site if you like.
2004 is quite late to be looking at UK web history since development started to take off in 1995. Of course The Internet Archive aka The Way Back Machine goes back to about 1996.
I just had a look on there for some of the sites we set up back in 1995. They used to be on there from the start (wasn't a huge amount to archive back then) with all their pages. UK Index is on there as at 1996, but only has one page indexed. A pity since that site used to host the very first National Trust website. Our earliest site Emoticon's earliest entry is now 2000,the earlier versions used to be there for all to see, but no longer.
Of course it is difficult to archive websites. They are continually changing (as an aside Sightsavers International have just unveiled their new design to celebrate their 60th Anniversary, take a look - very nice) especially with social interaction we see to day.
So, read about the British Library Web Collections and take a look (http://www.webarchive.org.uk/ukwa/) . As by next year they hope to have have archived just 6,000 of an estimated 8m sites, or 1% of the total if you want to ensure that your site (or its versions) is stored for posterity then it's probably up to you to do it yourself.
Out of interest, which UK sites would you like to see archived?