Talk to your boss immediately and begin job searching now
You have just learned a very important, and unflattering fact about your new company. Namely, they don't want to hire you right now.
I know you're excited about the new position, but take a long hard look at your new company. This is probably a place you DO NOT want to work. The rest of this post assumes you will not start working there.
First, talk to your boss, and try to save your current job say something like.
I know I turned in my resignation, but my new company just told me they can't hire me right now. Would it be possible to rescind/cancel my resignation? I have no intention of working for [new company].
Be apologetic, realize that your boss may decide that you'll be out the door soon anyway, and not allow you to "un-resign". Remember, they may have already reached out to another person to replace you.
It's probably worth specifically mentioning you don't want to work for New Company anymore.
Continue job searching
You've just announced to your current company you're heading for the door. They may look to lay you off or fire you in the near future. Unless you know this isn't true, keep job searching (be really sure about this).
If you're without a paycheck, then go work for New Company as a stop-gap, but don't stay long term. They will likely find new ways to not pay you.