Earlier quoted context omitted.
You could just use KeePass: https://keepass.info/ It's a free open source app that runs on your local machine and stores your passwords locally - never uploads your passwords to a server. But it does this securely. And you can run it on multiple machines (and phones) and transfer the passwords (the vault) without ever uploading anything to servers.
Thanks for the tip! I will look into it. I am curious to find out how it syncs without servers. (I assume this is not an incredibly hard problem but we are just not used to doing things without "the cloud" these days)
Obviously it's not as nice as having a cloud service, but it's open-source and doesn't require trusting a third party, which I like.