I haven't seen this one in Delhi for at least two decades, but it might exist in the nooks and crannies of that city that are unknown to me.
In Mumbai, however, at least in the large residential area where I stay, there are three or four such libraries, and only one of these deals in books and movies. The other two are solely books--English, Hindi and Marathi.
One of them is the size of a kirana shop in the less affluent parts of Delhi. It's managed by three women--who take turns to come and sit there, aided by a young boy who can climb up to the top shelves to retrieve books. The monthly rates are nominal, less than what one brand new bestseller would cost. There are no fines and no limit on how many books or magazines you can read in a month.
In the surrounding glitz--there's a row of boutiques three shops down, three eateries, toy shops and grocery stores, and even a jewellery showroom in the vicinity--this place comes as a surprise. It's housed among a shop selling lights, a doctor's clinic, and a vegetable vendor, which is strategic positioning--people can visit it on their daily grocery shopping trips, or while returning from office. The bigger surprise is that the library seems to be doing well on the usual fare of English bestsellers, Hindi classics, Marathi novels, children's books and loads of magazines in all three languages. I guess just like roti and rice, some reading is also a staple in every diet.