Both of these issues can be largely mitigated by simply plugging the HDDs in every year or so and re-writing the data (maybe run badblocks while you're at it to ensure the drive is still in good health), but, as throw0101a mentioned, this takes time and effort.
[0]: https://serverfault.com/a/51893
[1]: https://www.partitionwizard.com/clone-disk/do-hard-drives-go...
How long a given SSD will last before losing data depends on the chips it is built with. I remember reading that Sandisk rates their flash storage as being able to survive a year without power (though I read that years ago and cannot find a source for it now), though [0] says 2-5 years, so technology has likely advanced since I last read about this. Either way, you should probably try and power up your SSDs at least once a year to keep the cells charged. This is less work than to keep an HDD healthy for cold storage, but it certainly isn't "set and forget" like tapes or optical disks.
Edit: I remember pulling my first SSD, a 60 GB Intel 330, from an old laptop and, despite having not been powered on for 3 years, it still booted. But that was MLC/DLC [1], which only has 4 voltage states, and I only saw that it booted, not that all my old data was intact.
[0]: https://www.easeus.com/resource/does-ssd-need-power.html
[1]: https://www.intel.com/content/dam/www/public/us/en/documents...
Manual labour.
There are tape libraries with robots that can take a tape out of a slot, put it in a drive, and then put it back automatically without humans needing to do anything. If you want to move the tapes offsite, there is some manual work to move the tapes from the import/export slot to a container.
A large library like the StorageTek SL8500 can have up to 640 drives, 100'000 tape storage slots, and 2'880 import/export slots.
Big cloud providers have disk replacement robots, so it's not unfeasible for robots to perform the "unplug and store the HDD somewhere".
But another advantage tape has is durability.
[citation needed]
I have no idea if anything like this is actually in use, I have my doubts. Also, a system like this probably isn't terribly useful, a Cartesian gantry robot would be much faster and way more robust than an articulated arm strapped to a mobile robot platform.
Wow the scale is just immense, that's pretty cool!
So, yeah, spinning rust might rust.