Wow. Looks like a giant pain in the ass.
According to the Social Security website, you need at least a birth certificate. If you don't have one, you have to go to your local Office of Vital Statistics and apply for a "delayed birth certificate." Looks like the requirements for such vary from state to state, but according to, say, the Kansas rules, you're going to need at least two documents stating your true DOB and birthplace (list of acceptable dox on p. 2), in addition to two notarized affidavits from attending physician, midwife, parents or older relatives confirming as much. So that kind of answers your birth certificate question.
If you can't get that, it looks like the SSA would also accept a religious record from before the age of five confirming birth. But it needs to be the original copy, not a photocopy.
In the unlikely event that you did get a delayed birth certificate (or found an original copy of a religious record confirming your birth off the grid), you'd also need to present the SSA with proof of identity, such as a driver's license, passport, etc. But given that pretty much every state requires you to provide... a birth certificate and a Social Security card to get such an I.D., that option isn't really viable. In absence of those, you could provide and employee identification card, school identification card, health insurance card or military I.D. card. I'm not sure how many of those you could produce if you never had an SSN, though- don't think I've ever worked a job that provided me with I.D. that didn't also require all the requisite pre-hire documentation.
Ooo, here's a sheet that looks like it offers some easier, more promising options.
Honestly? I have no idea how you'd do any of this without forging something. Might be easier just to pay somebody shady to make you fake papers.
Edit Here's an article that addresses it a bit: