There are a lot of places in the world that are considered backpacking destinations. It seems as though the only thing that a place needs to qualify as a backpacking destination is to have cheaper than hotel accommodation that includes shared facilities. I was very surprised when an international friend of mine was able to find hostels with shared dorms in New York City. I had no idea such a thing existed. He was able to get a bunk for around $25 a night in a rather busy part of the city and near the trains.
In my mind, this doesn't really make New York qualify as a backpacking destination though because as soon as you step out the door of the hostel, you are still in one of the most expensive cities in the world. I have a few friends that live in the Big Apple and unfortunately I very rarely visit them because in just a few days you will tear through a couple of weeks' budget just by living life even if you are trying to be frugal. Things are just really expensive in the world's most busy cities.
The "HI!" New York City Hostel is a great value... unfortunately the rest of the city is NOT
The same person that I visit in NYC also went "backpacking" kind of. He went to Europe and bought a train pass and went to a lot of the sights that this famous continent offers but lets just say that his financial experience was a lot different than mine has been because I went a different route.
A lot of the countries that I have traveled to are in relatively poor parts of the world. These locations include many Central and South American countries, Nepal, India, and basically all of South East Asia. My budget in all of these places could be as low as $10 a day (this was 20 years ago) and these days it is still possible to visit almost all of these places on a budget of $20 a day and this includes a few drinks every day as well.
There is no chance that you are going to even be able to score a bed in most European countries on this budget.
This isn't to say that I don't consider traveling around Europe to be true backpacking, I do consider it as such as well as someone who turns up to NYC to look for bargains. It is still dramatically cheaper than staying in a hotel and doing tours and what not and it certainly is a lot more adventurous that going on a heavily traveled road. However, it is easily 5x the price to do so and that is why I haven't done it yet.
You wont find very many people that are backpacking for a year in Europe, Australia, or New Zealand for example, because it simply is out of the reach financially speaking, for most people.
I found this chart of the average cost of backpacking major cities in Europe and to me, this is really expensive when we start talking about budget travel. Prague, I have heard, is an anomaly in Europe and this is represented above. For the most part I NEVER spent this much per day (aside from the Prague range) in almost any city that I have stayed in in the dozens of countries that I recommend to backpackers. I think that if I ever spent more than $50 in a day it was either a travel day involving planes, or a big blowout party with friends for some sort of special occasion.
So while I still encourage people to travel to these sorts of places, I'm afraid that it is still out of reach for me, financially speaking. In my experiences I have much moreso enjoyed being in a country where the prices were cheap for everything and your finances were therefore something you didn't really need to stress about.
Will I ever travel to these places? To be honest, probably not. When I know that there is so much world for me to see for a fraction of the price I think that I would rather stick to the cheaper countries. If I ever travel to Europe the Western countries and Scandinavia will likely have to be eliminated from the list and I will simply stick the the Eastern part of the continent.
If you have experienced differently please tell me about it. I can't really comment on the things I haven't seen but in my travels I am yet to meet a single person that can regale me with their stories of inexpensive long-term stays in Europe.