Archive.org – How to Archive a Whole Website

As their FAQ's state:

How can I get my site included in the Wayback Machine?

Much of our archived web data comes from our own crawls or from Alexa Internet's crawls. Neither organization has a "crawl my site now!" submission process. Internet Archive's crawls tend to find sites that are well linked from other sites. The best way to ensure that we find your web site is to make sure it is included in online directories and that similar/related sites link to you.

Alexa Internet uses its own methods to discover sites to crawl. It may be helpful to install the free Alexa toolbar and visit the site you want crawled to make sure they know about it.

Regardless of who is crawling the site, you should ensure that your site's 'robots.txt' rules and in-page META robots directives do not tell crawlers to avoid your site.

When a site is crawled, there is usually at least a 6-month lag, and sometimes as much as a 24-month lag, between the date that web pages are crawled and when they appear in the Wayback Machine.

In some cases, crawled content from certain projects may appear in a much shorter timeframe — as little as a few weeks from when it was crawled. Older material for the same pages and sites may still appear separately, months later.

Why isn't the site I'm looking for in the archive?

Some sites may not be included because the automated crawlers were unaware of their existence at the time of the crawl. It's also possible that some sites were not archived because they were password protected, blocked by robots.txt, or otherwise inaccessible to our automated systems. Siteowners might have also requested that their sites be excluded from the Wayback Machine. When this has occurred, you will see a "blocked site error" message. When a site is excluded because of robots.txt you will see a "robots.txt query exclusion error" message.

The exclusions are at the choice of siteowners or due to take-down notices. I'm afraid if you are asking whether you can decide to include what is presently excluded the answer is that you cannot (for other than your own site/s).