I was wrong. You may actually be able to delete your old embarrassing MySpace profile even if you're completely locked out of the account. But you must pose for a picture with your MySpace URL like it's 2003 and you're trying to prove to your webcam girlfriend that you're not a total scammer.
Do you want to shut down your old MySpace profile but have forgotten your password and no longer have access to the email address you used to open your account? (And you should!) Stefan Becket points out this MySpace support video, which explains how to shut down your account in this case.
1) To prove your identity, you must take a picture of yourself doing what the video calls a "MySpace salute." This is not duckfacing with your two bffs in a mirror, but a clear picture of your face posing with a handwritten sign displaying your MySpace URL. (I have included a topless picture of MySpace investor Justin Timberlake. After all the layoffs, he probably handles these requests personally.)
2) Fill out a Myspace contact form here. Explain in the body that you want to delete your MySpace profile.
3) Attach the picture of your MySpace Salute and submit the form.
This doesn't guarantee anything. MySpace has been doing the salute thing for years, and Google is littered with people complaining that sending in a MySpace Salute didn't get their account deleted. (Here's a great gallery of forlorn users doing the MySpace salute.) If it doesn't work, you can also try this Declaration of Ownership form. Who knows.
In conclusion: MySpace, please shut down my goddamn account.
Note: Even if you can log into your MySpace account to deactivate it from the inside, you still need access to the email address you originally used to register it in order to confirm the deactivation. If you don't, you'll need to do the Myspace salute.