If you're using this on an iPhone, the failure is quite non-informative. Trying to browse to a URL looks like you failed to input anything and it just falls to a blank screen.
On the Mac, you'll see the message "Safari can’t open the page. The error is: “The operation couldn’t be completed. No such file or directory” (NSPOSIXErrorDomain:2)".
On iOS there was a message "iCloud Private Relay will be available tomorrow" - and things seemed to keep working for me at least.
On macOS it took 16 hours (!!) before a message finally popped up that said "Private Relay Unavailable - Private relay will automatically turn back on when it is available."
On the Mac, you'll see the message "Safari can’t open the page. The error is: “The operation couldn’t be completed. No such file or directory” (NSPOSIXErrorDomain:2)".