I can't sign in to Xbox Live Windows 10 (0x405)?

I can access the Windows Store without any problems, but as soon as I try to log in to the Xbox Console Companion app or the new Xbox app, I have "no network connection". I have perfect hardware, the latest Windows update, a LAN cable and an otherwise working Xbox account.

I don't know what's going on.

Login is grayed out in the new app

Here at least the field is there but:

First it loads

and then I end up here, because the app doesn't work, I can't get any further

Just reinstall the apps. If that doesn't work, I recommend the Xbox Game Bar. It's implemented much better than the other two apps. (Key combination = Windows key + G)