Is it illegal for me to create a backup of a website that is shutting down, including all assets, etc., and host it on my website?
As the content on the site is copyrighted, and as any logos may be trademarked, it would be illegal to host their content without their permission.
Of course, it may be the case that the site is closing down because it doesn't want to pay for the costs of keeping it going, as such it may something the owners of the site would appreciate if you offer to host their site for free, indefinitely.
The site is copyrighted so yes, obviously you can not copy it. You can contact them and ask for whatever content you want though.