I tried using 2020 AGI of -116 (which is on my 2020 return) and also 0, both were rejected.
And since Federal is rejected, NY state return is also rejected. But NY state requires efile so I can't even mail in.
You should be fine filing your New York return on paper.
According to the state’s E-file requirement for individual taxpayers page:
You're required to electronically file your return if you meet all three of the following conditions:
I suggest filing your state return by mail and attaching a letter stating what you’ve said here – that you weren’t able to e-file your federal return, so you can’t e-file your New York return. And it wouldn’t do you any harm to actually print this Community page, with the URL visible, and attach it to your return, as well. That would show that you’ve done your due diligence – you’ve tried to e-file, but simply can’t.