In the time before the Internet where everyone got their news from published newspapers, it would be much harder for "news" and "history" to get deleted and re-written / censored like how it's happening now with MSM and Big Tech deleting posts of news that they don't like. Alternatively, back in those days, it could also be done by burning books and newspapers, but I figure it would be a lot harder.
Is there something like an Archive.org that could save all the posts deleted by big tech and MSM?
You should use
https://archive.today.
It takes an active request by users to snapshot the webpage, but once captured, it's on there for good.
Archive.org is actually the "wayback machine" site. It's been known to "lose" snapshots users had created, so don't trust it as an archive.
In the "old days" of the Web, you could use a mirroring app to copy a page, a branch, or an entire website to your local storage. It's used to migrate or replicate websites or pages so you get all the files associated with a complete page/site.
Nowadays, the majority of webpages do not exist as a webpage. They are composed of data (usually in a database & in "flat" files), templates, and software (normally PHP or other interpreted language), which is then presented to the user's client (browser) in a webpage format (HTML). The page you see didn't exist when you requested it, was built on-the-fly to be presented to you, and will cease to exist when you leave that URL/IRL. This makes it much more difficult to capture a webpage unless you are currently viewing it. This is why you need to request a snapshot be saved to any storage location, local or "in the cloud".
Realistically, you could also use your computer's image capture apps to do this. Or you can even use a camera/cell phone and just take a snapshot that way.
Unfortunately, for you to be able to capture everything you might want to use later is impossible. How do you know what to capture until you need it?
People posting YT videos and referencing articles often do captures of the sources they are referencing, because one of the most dishonest things some publishers do is replace one article full of lies with a different one at the same link. That avoids the appearance that anything was deleted -- just not as the content creator on YT had seen it. Lots of that went on in the 2016 election. By the time most people got the link, a negative Clinton article was changed to a piece on how bad the Orange Man is. Bait and switch? or just censorship?