I can't even imagine how rough it must be on some people who constantly have to deal with the rejections. I still urge you not to lose hope over your situation and especially do not find fault within yourself. The legal job market for entry level is shit, I have no other description for it. Also, many of those who have been in the profession longer simply do not understand the difficulties you're facing. Do you really think a lot of older attorneys have any idea how much law school tuition is these days? Unless they're about to put a kid through college they won't have the slightest clue. Older lawyers are simply very removed from the issues a younger attorney faces at the start of his or her career.
Why am I telling you this? I am not sure, because I do understand that it is not really helpful. I don't have a lot of advice to give, either. I believe I have said it before, but you really need to sit down and make an inventory of those people in the profession you know and who you can reach out to. Not for a job, of course.
But think about supervisors, professors, attorneys you worked with, judges you interned for and law school colleagues who might have lucked out and found a job already. Think about people your parents might know, think about people in the community you know who know others. Politicians, preachers, what have you. Get in touch with those people and ask them for advice and let them know about your situation. You need to make them understand how desperate your situation is. That takes some courage and some diplomacy, but many of those people will be willing to help.
While they might not be able to offer you a position it will already be helpful if they are willing to keep a couple of your resumes in a drawer in case they hear about something somewhere. Networking by yourself and cold calling is extremely difficult and, in my opinion, pretty useless for starting attorneys. But if you can manage to get a few people who keep an eye out for you, that will be helpful and eventually pay off. You need champions who will recommend to somebody looking that you're the person they should call.
So keep your chin up, people. It can be done. You went through all this crap to get where you are now and you are now entitled to join this profession. Don't give up now and don't become cynical.
I have faith in you. You wanted to be damn lawyers so go ahead and be your own first client to advocate for.