Is it possible to have gotten by without one?
Say someone didn’t get one, and worked under the table many years and , obviously, never went to a doctor for anything.
Would they be an invisible citizen?
Allowed to vote?
Arrested?
Well, mere lack of a SS# won’t cause you to be arrested. Nor will it deny you the right to vote (I’ve never had to show my SS card or give my number when I registered.)
However, since there is an incentive for parents to get SS#s for their kids (to claim them as tax-deductions, the kids must have SS#), it is unlikely that anyone who grew up in the U.S. in the last 20 years doesn’t have a SS#.
Zev Steinhardt
When i had my son, they wouldn’t give me his birth cretifictae until I had gotten a s.s. number (or sent away for one)!
Yeah, me too. The SS lady at the hospital was downright freak-a-zoid about getting Kate a SS#.
I don’t think that a tax incentive matters if you don’t have a SS# and don’t pay taxes. 
Is it possible? Well, since everybody got by without one prior to 1935, I suppose it’s possible.
Today, however, it’d be tough. You’d not only have to be paid your wages “under the table” and forgo the usual brand of medical care, you’d also have to forget about ever getting a driver’s license or legally owning an automobile, having a bank account, obtaining insurance, serving in the military (if you’re so inclined), attending college, etc. etc… Even most utility companies these days ask for an SSN.
Of course, on the up side, no SSN makes it possible (but definitely not legal) to avoid paying income taxes. Of course, if and when the IRS comes calling, you’re looking at the “arrested” part…
Without social sercurity number one cannot pay taxes (aw, shucks), cannot obtain any time of government assistance, cannot establish a credit rating (no mortgage, no car, no credit cards), cannot have a bank account, yadda, yadda.
Is it possible? Maybe. Is it smart? Well, for all intents and purposes you would be a homeless, non-person in your own country. You might get by for a number of years if you stay healthy. But age and age-related illnesses would do you in.