Sure, but what should I do when my all digital storage (including cloud one) fails?
Make more backups in other locations so it's unlikely that everything would fail simultaneously. And when a storage device fails, the data isn't gone permanently unless the device is physically destroyed (but you should have remote backups), and even then, not always. It's possible to recover data from storage devices even after they are "dead". For example, if you use a spinning hard disk drive and the drive motor, control board, or read head fails (some of the most common causes of drive failures), specialized data recovery services can open up that drive, pull out the platters, and recover all of your data anyways. And you can avoid catastrophic failures by using multiple drives in a RAID array.
Obviously you should have remote backups, whether that's cloud storage or a physical drive you plug in every so often and put a backup on it.
Bit rot is a thing that could affect your backups, but routinely making and verifying backups is something that you should be doing so it really isn't a problem.
If you insist on having a paper backup, you can hex encode or base64 encode the wallet.dat file and print that out
I suppose what you are really looking for is the HD seed that Bitcoin Core uses. The HD seed is used to derive the xprv which is then used to derive the child keys. The HD seed is what Bitcoin Core is able to store and use, not the derived xprv. You can find the HD seed in the file created by
dumpwallet. It is a private key that is marked with
hdseed=1. You'll probably also want any of your previous HD seeds (if you've changed HD seed before or just encrypted your wallet but had used it before encrypting). Those are private keys marked with
inactivehdseed=1. All HD seeds will have
hdkeypath=s so that's another way you can find them.
A new Bitcoin Core wallet can have a HD seed set by using the
sethdseed command.
But, I still wouldn't recommend backing up the HD seed. This kind of backup, while it does backup your private keys, loses a ton of other data. You'll lose all transaction data, which may be annoying to get back if you've exhausted the keypool before (in general, keypools and gap limits are issues with private key only backups). You'll lose any metadata like labels. And you lose the creation date and time of keys (important for making rescans faster).
What you really should be doing for backups depends on what your thread model is. What are you trying to protect against by backing up?
Are you trying to protect against ransomware or other malware that locks you out of your computer? An offline digital backup is sufficient. You could use a hard drive that you backup your whole PC to every once in a while. Or use a remote storage service. You should also encrypt it. To protect against that case, you should be backing up your entire PC anyways, so making an additional physical backup to protect against it doesn't make that much sense.
Are you trying to protect against your house burning down (or some natural disaster) destroying everything in it? Well, your physical paper backup isn't going to stand against a fire. So in that case, again, a digital backup to a remote storage service. You could use a sturdier physical storage medium like metal, but good luck finding it after your house burns down. But it could still be destroyed or rendered unusable by damage. Yes, you could send that physical backup to somewhere else to be stored, but then you need to trust that whoever you are storing it with does so securely, and that that place isn't also destroyed. At least with a remote digital backup, you can be certain that the service provider is backing up their stuff so if their data center is destroyed, your data will still accessible. And you can encrypt it so it will be secure.
Or are you one of those people who uses tails and then nukes the computer they used it on (and more generally, uses ephemeral OSes so everything needs to be re-installed and re-initialized every use)? In that case, a physical backup is probably better for you. Obviously you want to keep your private keys as offline as possible, only putting them on a computer for as little time as necessary. So a physical offline backup is better for your use case. But you still need to consider the implications of your backup being lost or destroyed, so you better have multiple and you'll need to trust whoever is storing them for you.
So suppose you've made digital backups and a physical backup, and the reason for the physical backup is "just in case all digital backups fail simultaneously". I don't think that's a good reason to make a physical backup. The reason being, if all digital backups failed simultaneously, you've got bigger things to worry about. Think about what might cause your own computer to physically fail and data become unrecoverable. Think about what a remote service who specializes in cloud storage to lose all of their data and all of their backups. For both to happen simultaneously, the only things I can think of are cataclysmic events. Either some terrible natural disaster has occurred like an asteroid hit the Earth, or a war has broken out and a bunch of cities just got nuked. In either case, you are either dead or have bigger things to worry about than the safety of your Bitcoin. Hell, money probably doesn't matter, and the Internet will basically be destroyed (tons of physical connections will have been severed).
So why would making a physical backup be a bad idea? Well, it's another thing to keep track of, another thing to verify still works. And most people aren't going to encrypt their physical backups (because the UX for doing that sucks). So if you lose that backup or it gets stolen, you'll need to go through the hassle of making a new wallet and moving all of your coins there. Otherwise you risk your coins being stolen. So why risk that? Just make a digital backup instead.
In general, I don't really recommend using paper backups if you are using just a software wallet. There are so many other things in a wallet that are necessary for working, more than just private keys. The wallet file provides those things, so really you should be backing up the wallet file, not just the private keys.
If you're using a hardware wallet though, then yes, making a paper offline backup makes sense. The seed is all you need to restore a hardware wallet, but is not enough to restore a software wallet, you need to have both the hardware wallet seed and the wallet file data to truly restore your wallet. But it doesn't make sense to digitally backup the hardware wallet's seed since the whole point of a hardware wallet is to have a kind of airgap. So a physical backup makes sense there. But, as mentioned earlier, you still need to protect against that backup being lost or destroyed.
