The English use the word “Proper” to signify something that is correct. That of course is their interpretation of what is correct, and can be quite annoying and judgmental. They can be real wanks.
But I would say that the Hostel that I am staying is a proper Hostel. It has a common kitchen, television room, book exchange and free breakfast in the morning. There is coffee and I think tea free all day. The kitchen has everything needed to cook.
The free breakfast, coffee, and tea is a bonus. I would not consider this necessary to be proper. The only thing lacking in this Hostel is a Locker that I can use with my own lock. There is the always in any Hostel or Hotel the danger of having things stolen still. Although I think the manager screens the residents and sort of decides whether they are safe. He is from South Africa and the Hostel has a good representation of countries. I just realized that there are no people from England here. But I am in England and normally the people from England would already have a home. There are mostly Australians and New Zealanders. But there is one more American and a Brazilian, a couple Germans and I am not sure of most of them.
I just walked down the street and found this place. This is really the first one I entered. Cost is 15 Pounds for one night, but cost 87 pounds for a week. (Note the date if you read this a year from now)