It honestly just depends. What you're asking for is less traditional hacking and more forensic recovery though. But it 100% depends on a wide variety of factors. As for how much it'd cost, anyone that would have the skills, like myself, to do this, would obviously charge what they're worth. And I'm not on here to do anything illegal, just make friends. But for the amount of work that typically goes into this, it's not usually worth it for the client as I personally will charge hundreds - thousands of dollars depending on the task(s) at hand.
With that said, I can point you to some OSINT tools that might be able to help. These are publicly available tools that use legal data gathering techniques! First, I'd check on https://haveibeenpwned.com/ and see if any of those old emails have been leaked which there's a good chance they might! Also, if you remember any of the usernames you can use software like sherlock.py to try and track some of the ones you might have forgotten about! From there it would really just depend on the website and email host, depending on what you're looking for, on how hard and, if possible, data recovery would be.
If, say, this was a kongregate account from 2013, you may get lucky and someone's leaked a database with their info, it does happen. But if you wanted to say, hack someone's gmail, it's a very involved process that's a pain in the buttox and would, if possible, take someone like me awhile because now we're doing something that's targeted rather than just going for what's easy peasy. Exploits exist but they're not shared around, then they wouldn't be exploits anymore! But for that type of thing is why I'd charge hundreds - thousands of dollars. But I'm more than happy to offer advice and things to try in your recovery efforts!