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Get wayback machine proxy software - https://github.com/STRML/wayback-machine-machine 
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Configure proxy software, and setup proxy on your OS. Run proxy software. 
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If using Windows, get WGet software - https://eternallybored.org/misc/wget/ 
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Open a command/terminal window in the directory you want to download the site to, use this command: wget -r -np -e use_proxy=yes -e http_proxy=127.0.0.1:4080 -k http://www.mywaybackmachinewebsite.com
NOTE: If you want, visit the website first to make sure it's exactly what you want prior to download. WGet will download whatever your browser sees.