The Artemis Yoruk was a short walk from the springs and was a cozy little place. There is a nice pool in the courtyard. I only had two complaints: first, the wifi is quite unreliable, perfect at times and then unusable a whole morning. The other issue was that when I asked about buses the owner was only willing to help me arrange for one of their private buses to my next city. It wasn't an unreasonable price, but I was turned off by his unwillingness to discuss the public bus.
The CG was very clean and the dorm rooms were colorful and well decorated. It was right next to the old city and just across the street from a bus stop. I was the only one staying there, so there wasn't anyone to spend time with, but I assume that was a fluke.
The New Swedish Hostel is in a great location in the old city and it was clean enough, but the staff wasn't particularly knowledgeable about the nearby tourist attractions and one man in particular was downright rude. There was no common area, so it was very hard to get to know other travelers. This place is perfectly adequate in a pinch, but I wouldn't want to stay more than one or two nights.
The Sabaa is a short 10 minute walk from Petra, or you can easily get a cab for 1-2 JD. It was very clean with a roof with great views of Wadi Musa. Gail, the owner, is a wonderful resource for ideas about how to use your time in Petra. Highly recommended!
My friend and I had an 8 bed room to ourselves, the only other guests seemed to be two large groups of middle school students! Not a great place to meet other travelers but it was immaculate and a short walk from the docks and the Cavern Club. Breakfast is pricey but there is a nice self-service kitchen to use as an alternate.
Very cute hostel, affordable and clean. The food is very blah, but there are good options in town. You can rent snorkel gear for 6 JD and the beach is right across the street.