You won't get citizenship unless you have legal work, which requires a DNI. So, yes you can stay here illegally. And, this will meet the requirements of "residency". However, you will not get citizenship without legal work here. So, there is not much reason to stay here illegally if you want citizenship because without the legal work you will not get citizenship.
If you were to have an argentine live-in girlfriend who did have legal work and supported you, then maybe you could get around that. Otherwise there is no way you will get it with the current climate. It is better to look at a country where you can qualify for residency and pursue citizenship that way. I would recommend Paraguay.