I've already overstayed my 90 day period as well as the grace period so I'm officially illegal...I'm working full-time but the company is based in Canada (I.e they don't provide anything in terms of visa sponsorship/help here in Arg) and renting an apartment for myself, obviously paying in cash.
What steps should I take and in what order? What should I do first, second, etc in order to start the process of obtaining some type of legal residency while giving myself the least chance of being deported/detained on the spot wherever I go..
As Dr. Rubilar indicated, you could apply for citizenship, but (based on his posts in other threads) you may have to have already been living in Argentina about a year to begin the process. Though I don't have the latest figures, I believe that you would also have to spend at least $6000 USD to get the citizenship. The fee varies from one "citizenship attorney" to another. Dr. Rubilar has been the trailblazer in obtaining citizenship for foreigners without legal residency.
If I remember correctly, a member from Canada once posted that Canada does not "allow" dual citizenship. Dr, Rubilar recently posted that you do not have to renounce foreign citizenship when you are granted citizenship in Argentina, so it might be "possible" to have both.
Others have posted that the "easiest" temporary residency to get is as a student, so that might be a possibility, but you would have to actually enroll in a "school" and attend classes. I don't know if taking Spanish or Tango lessons would get you a student visa.
Neither do I know (since the decree DNU 70/2017 went into effect) how aggressive migraciones has been about hunting down foreigners who have overstayed their 90 day visas. If you provided the address where you are now renting on the immigration form when you first arrived in Argentina, you will be easier to find.than if you found and rented your apartment after your arrival.
You and Orion are in the same situation as far as haveing overstayed your visas as well as the grace period. You both are "honest people" as Dr, Rubilar uses the expression, but the DNU 70/2107 appears to target foreigners whose only "offense" is not having valid residency..
As a result of the decree, foreigners in your situation no longer have the rights that are (in Dr. Rubilar's esteemed opinion) guaranteed by the Argentine Constitution. If I remember correctly he recently posted that the enforcement of the decree had become less aggressive, but I wonder to what extent the federal police are cooperating with migraciones and what actions they might take if you are involved in an accident or make a report following a robbery or mugging.
Based on the factors I've mentioned, as well as some that may be "unknown" at this point, it's impossible to say how long you will be able to stay under the radar.
I'm curious to know (prior to your arrival in Argentina) if you were aware of the residency requirements for staying in Argentina beyond the expiration of your 90 day visa and the one time 90 day extension that was available for up to 30 days after it expired (if you had asked for it). It might also be helpful to others who may read this topic in the future if your employer "sent"you to work in Argentina without legal residency and if they were aware of the possible consequences of working here without it.
If you were working for a company that was registered to do business in Argentina I believe that you would be "paying"income tax on your salary that exceeds (approximately) $20K pesos per month and your employer would be paying at least something toward your health insurance (obrera social?). and your retirement benefits.
What will you do if you. get sick or need health care? Do you have private insurance? I've had uninterrupted "legal:" residency in Argentina for over eleven years, so I don't know if it's possible to get private coverage without it. I don't think a public hospital would deny you care without a valid visa, but seeking care at a private hospital might be a wise alternative.
I suggest you contact your employer and make sure that they know the details (and possible consequences) of your current immigration status. If migraciones knows your current address I suggest you make arrangements with someone to bring your personal stuff to you (at the airport?) on the day of your deportation if you are not allowed to return to the apartment prior to your departure. You will have some time after you are detained (at least two weeks?) to contest the deportation order, but if you do, you will be detained much longer.
Of course the easiest "solution" at this point is to buy a ticket back to Canada, pack your stuff, pay the overstay fine at the airport, and leave Argentina. If you do that I think you will be able to return to Argentina in the future, but I have no idea how long you should wait before doing so.If you wait to be deported I doubt if you will be able to return any time soon. It might be five years. Dr. Rubilar will know for sure.
In another thread Dr. Rubilar indicated he believed the purpose (or at least one of them) of the DNU 17/1917 was "ethnic cleansing" of non whites, but it appears to me that foreigners (regardless of race) who are working in Argentina without legal residency are also a target of the decree,
Unless you find a category of residency you can qualify for (and are sure you'll be approved when you apply for it) or you are willing to apply for citizenship (and have been in Argentina long enough to start the process), I believe you should seriously consider leaving of your own volition...while you still can.
If I'm wrong about any of this I'm sure Dr. Rubilar will correct me. He is always welcome to do so.