Never rely on any cloud service! A good cloud based password manager is end to end encrypted meaning the password manager provider cannot access your passwords and they are secured from the provider and any compromise of the provider. But you do not only need confidentiality but also reliability. The cloud is just someone else’s computer that you store your data on. They can cease their service or stop providing you access to it at any time. Always have a local backup of anything important saved in a cloud.
With Bitwarden for example you can export your vault as unencrypted json and csv format. Those are widely compatible and allow you to easily access and import your passwords.
Do not save your exported passwords unencrypted. I strongly recommend creating a dedicated VeraCrypt or LUKS container or similar and saving the export directly into that without saving it to disk unencrypted in the first place.
Note that shared organizations are not included in the standard vault export and need to be exported separately.
Edit: Someone mentioned that Bitwarden’s export feature does not export attachments. So export them manually if you need to.
Good tip, I’ll get on it. I use Bitwarden’s cloud server for convenience, but if they were to shut down tomorrow, I’d be screwed
Only somewhat screwed. The client still has a local offline cache. So you don’t immediately lose everything. The cache is read-only though and doesn’t contain attachments.
Also remember: the normal export function of bitwarden also “just” exports the database entries; not the file attachments.
If you are the BitWarden server admin, what do you see per user? Just a blob of data called “encrypted password database” or something, along with something like a last modified date and version number for syncing purposes?