I think that if you are travelling alone, a hostel is much better than renting a place. Besides, apartments are mostly located at the expensive neighbourhoods.
It won't be easy to find a job, there are lots of laws protecting employees, so you might need permission to sign a contract. But don't worry, since 2001, when 1 dollar became more than 3 pesos, Argentina turned into a very cheap place to visit.
As regards language, whenever you stick to tourist activities, many people speak English. Shops, restaurants and bars are used to serve foreign customers, so you'll be ok.
I believe hostels don't have gyms. They are quite expensive, but you can try at "Facultad de Derecho" (Law School) or "Los bosques de Palermo" (race track, very nice).
Just one piece of advice, be very carefull. There are great people, kind and helpful, but also lots of thieves. I'm Argentinian, from the South, studied in Buenos Aires and had to be extremely careful until I learnt which places were safe and which had to be avoided. Now I'm living in Rio de Janeiro, one of the most dangerous cities ever, and I'm doing the same thing. Be cautious, keep yourself informed, find the busiest streets always, avoid wearing gold, hide your camera, wallet. Choose daylight.
Anyway Pete, you'll surely have a great time in that splendid country.
Now I'm working in a travel agency, learning, and I love to read comments and questions to see what interests people have.
There's a hostel that might be interesting (I found in the internet):
End of the World-San Telmo
Av. San Juan 818
Start your adventure at the END OF THE WORLD!! Internet Free 24 hrs. WI-FI Zone, Free Spanish Classes, Free Tango Lessons, Breakfast Included. Located at the heart of San Telmo, next to the Arts and Crafts fair, museum and Night Club area. Constitucion Station is only 3 blocks away.