Basically a labyrnth of bureaucracy, but can be done...
For those of you who are wishing to stay for an extended period of time in Argentina, or are eligible for a permanent residence, you will have to translate ALL of your necessary documents into Spanish- that includes yur passport! If you are eligible for a DNI, or national document (i.e. if you are married to an Argentine, or have Argentine parents), that could mean various trips to the "registro" where foreigners are sent. I had been told conflicting things, therefore had to go there about ten times before completing the "tramite".
They are sticklers for details, so the translated documents must be stamped in the proper way- I had a translated marriage certificate rejected because the translators seal was not stamped as is should be. That brings me to another point- they will not accept translations done abroad. They must be done by someone from the "Colegio Nacional de Traductores" in Argentina. You can easily obtain a list of the qualified translators. I hadn't known this, and wasted money on translations done in the U.S. before I arrived.
Depending on your case, they might ask for different documention, but be sure that you will have to get your passport, birth certificate, marriage certificate (if applicable), letter from your hometown police department stating that you don't have a criminal record, certificate of good health- all translated into Spanish. You might be better of going to a private doctor here, for the health certificate. I believe you can get the forms from your local argentinean consulate before you arrive here. I saw a doctor (recommended by the consulate) in the U.S. that was able to complete the form in Spanish. You also must have two photos, but not U.S. passport type- they must be slightly profile, the argentinean way. You can get the photos taken in the same building where you apply for residence.
I may have even left out some details, but this is an overview of what to expect. They can and will reject papers for any reason they can find. That doesn't mean you won't get your residency, just that you have to be patient, return, and comply with what they require. You do have to have a visa from the consulate abroad in order to start the process. You cannot come here on vacation, then decide that you'd fancy living here, and simply stay on. Well, you could, but it wouldn't technically be legal!
Good luck to any of you in this process!