It is possible to find a little studio, of less than 20 sqm, for around 600€ - 700€ but you'll need to do lots of research and as soon as you find one, take it because lots of people look for budget accommodation, there are more clients than flats available. This concern long term rentals, unfurnished. Furnished flats are easier to find but cost more expensive or are smaller. You'll easily find former maid's rooms, at the last floor of a building, without lift and with toilets outside. More modern buildings have the maid's rooms at the ground floor or 1st floor or have a separated building, so when you rent, ask if it's a former maid's room or not and if it's at the main building.
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It is possible to find a little studio, of less than 20 sqm, for around 600€ - 700€ but you'll need to do lots of research and as soon as you find one, take it because lots of people look for budget accommodation, there are more clients than flats available. This concern long term rentals, unfurnished. Furnished flats are easier to find but cost more expensive or are smaller. You'll easily find former maid's rooms, at the last floor of a building, without lift and with toilets outside. More modern buildings have the maid's rooms at the ground floor or 1st floor or have a separated building, so when you rent, ask if it's a former maid's room or not and if it's at the main building.
http://www.seloger.com/
http://www.pap.fr/
I doubt it. It's an expensive city to live in.