Paris has private hostels popular with youth and students, at which you need no official hostel membership card.

Perhaps the best way to find a hostel in your price range is through Hostelbookers:

Young and Happy Hostel
This private hostel in an old four-story apartment building in the Latin Quarter has two-, three-, and four-bedded dorm rooms. Accommodations are spartan to say the least, but the mood is youthful and upbeat.
80 Rue Mouffetard, 75005 Paris (Métro: Monge). 60 beds. Closed daily 11 am-5 pm. Book Young and Happy Hostel, Paris, France with Hostelbookers

The 3 Ducks Hostel
Run by students, this basic hostelry is in southwestern Paris, in the 15e arrondissement. The spirit is free and easy, the beds simple and cheap. Take the Métro to the Félix Faure station, then walk along Rue du Commerce toward the church, which is the centerpiece of Place Etienne Pernet; or walk southwest along Rue du Commerce from the Commerce Métro station.
5 Place Etienne Pernet, 75015 Paris (Métro: Félix Faure or Commerce). 40 beds in one-, two-, three-, and four-bedded rooms. Closed 11 am-5pm. Book The 3 ducks Hostel, Paris, France with Hostelbookers.