It seems like a lot of web pages are disappearing from the internet these days. If you feel like taking on archiving duties for yourself, there are a variety of tools for doing so.
Image: Indiana Jones and the Raiders of the Lost Ark.
Submit Pages to The Wayback Machine
The Internet Archive is a non-profit digital library that attempts collect as much digital knowledge as possible, including a vast collection of web pages. The Internet Archive's Wayback Machine is the most obvious choice for looking up an old version of a site, but it's also worth nothing that you can help with the process by manually saving specific web pages, including all of the assets on that page.
To save a site, head to the Wayback Machine, then paste a URL into the box that says Save Page Now. You can also use the Wayback Machine's browser extensions (Chrome/Firefox) to save any pages you're currently looking at with a couple of clicks.
Manually Download Pages Yourself
The Wayback Machine is great for the public good, but if you want your own personal backup, you'll want to look into a few different options. The most obvious of those is to simply download a page or site yourself.
Archive.is is probably the simplest way to do so and has the bonus of saving a site online as well. Drop a URL into Archive.is, and when it's done rendering the page, you'll see a Download Zip option to download the whole thing for your own safe keeping. Archive.is will also host that backup much in the same way that The Internet Archive does.
Of course, if you're comfortable with the command line, Wget is the easiest way to download and mirror a site in bulk.
Screen Capture Pages
If downloading the raw HTML files of a site isn't necessary, then screen captures might do the trick. This is also useful when the layout or visual style of a page is important, because oftentimes archive sites can mess things up slightly.
There are a ton of screen capture tools out in the world, but a few are better than others. If you're a Google Drive user, the Save to Google Drive extension can instantly convert a page to a screenshot, then save it Drive. If you don't use Drive, Full Page Screen Capture does the same thing but saves to your local drive. Otherwise, a desktop tool like PicPick on Windows or Skitch on Mac will do the trick. Most web browsers also have an Export to PDF option as well, if you'd prefer to save a page that way.
Save Pages with Your Notes App's Web Clipper
If you use a modern notes app like Evernote, OneNote, Google Keep or Bear, then you have access to a web clipper that saves the entire contents of a page to your notes. Once you install the web clipper extension for your notes app you can save a site, and the contents, optionally including images, will get saved.