The following is a way that - seemingly - not even Apple Support knows about, I've found it myself after I got told that my account's situation is hopeless and I cannot get it back.
I forgot the answers to all of my security questions and did not have a rescue email. There is no way to reset the questions in that scenario, your only option is to get rid of the questions altogether - and you can do that by enabling two-factor authentication. The catch is that you must answer some security questions to enable it from Settings.
And here is the interesting part: if you log in with your Apple ID on a device (I did it with an iPhone 5 running iOS 10, but any iOS 10+/macOS 10.12+ device should work), log out and do that several times, you'll eventually get a popup to enable 2FA... and it will let you do that without answering a single security question! I got the popup after logging in and out just once, but your results may vary.
Note: if you have macOS, you can use this method without logging out from your main account on your device - just create a new user, log in, and you'll be asked to link your iCloud account. (thanks for the tip, kallax!)
Others have confirmed that this method still works in March 2021:
I had a very old Apple ID with remaining credit that was only used for iTunes and not yet activated for iCloud. I had already changed the password via the email address some time ago. But resetting the security questions was not possible anymore, because I could not answer enough questions for that. And after several attempts even this form was blocked for the meantime.
On an old iPhone 4s (iOS 9.3.6) the Apple ID was still stored under iTunes & App Store (I don't know if it is required to use a device which was already linked to this Apple ID). I also entered the same Apple ID under iCloud to activate iCloud for this Apple ID. I then entered my phone number under iCloud > Apple ID > Contact information (confirmed by code from SMS). Entering an additional email adress was at this stage not possible. Thereafter, under iCloud > Apple ID > Password & Security, the two-factor authentication could be enabled for the same phone number (eventually an unlock code has to be set prior under Settings). I had to confirm the activation via the code sent by SMS. Logging in to icloud.com was then possible again and I was able to update all account information.