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 (2FA). The catch is that you mostly have to answer some security questions to enable it from Settings (sometimes you can get lucky and enable 2FA by only using your password and a texted verification code, no security questions - Apple Support was surprised that this worked).
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+ 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.