Organize you "keepers" into an archive you can maintain. at ~6TB (edited "gb" typo, tx) thats not a big challenge to back up; multiple usb hard drives and regular schedule backups of that will serve. "Remote" is a distraction; put your data on media you own and keep a copy in a bank box or something if you feel the need for more off site redundancy.
Collecting and arranging the archive is the big job. you're just gonna have to bite down and start doing that, yourself: no one else is likely to know what needs saving or not. Set up a NAS or file share with a big HDD and start collecting files there.
You may find the old stuff fun to scrape up; for example how difficult is it to find a PATA interface today? that problem only gets harder. Motivation to get on the job now, rather than later; and to make the "archive maintenance" more of your everyday task list than to let it pile up.
I'm a fan of the "low tech" redundant backup, that is making an independent copy of the data for example once a year on a new hdd and put that in your parents attic.
It's not exactly matching OP's requirements, but my solution is to regularly rsync important stuff to a small home server, on that homeserver do a borg-backup of the backup folder once a week onto a second drive. First one is ssd, second spinning rust.
In ops case I think the biggest problem is sifting through all the data. I did this a few years ago with only 4 or 5 old drives and a dozen dvds and it was exhausting, but also occasionally pretty nostalgic when I found photos from college or old programs I wrote as a teen.
Collecting and arranging the archive is the big job. you're just gonna have to bite down and start doing that, yourself: no one else is likely to know what needs saving or not. Set up a NAS or file share with a big HDD and start collecting files there.
You may find the old stuff fun to scrape up; for example how difficult is it to find a PATA interface today? that problem only gets harder. Motivation to get on the job now, rather than later; and to make the "archive maintenance" more of your everyday task list than to let it pile up.