What is the closest software Linux has that functions the same as MacOS Time Machine?

You can take a look at this thread to see if anyone has approached the topic closer to what you want:

Personally, I don’t rely on snapshots. I keep my /home on a separate partition and keep a list of applications I install. If I ever need to reinstall, my /home is intact with all my data and the dot folders hold all my settings, so a reinstall just uses my old /home partition and I choose the same username when I reinstall. Then I reinstall all my apps from the list I keep. I can be up and running in under 10-20 minutes from a dead system with very little loss.

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