I stayed at Fresh Sheets at the beginning and end of my trip in Croatia and loved both stays. It's located away from all the bars and restaurants along Stradun so it's a quiet place. It's also close to both Buža bars, which is a must-see/do I'm Dubrovnik. Staff was helpful and friendly, and leaving your bags there before it after you check in/out was no problem. The only downside - and it isn't much of one - are the number of stairs you have to climb. Definitely some good exercise!
Great location, not much by the way of a common room but who cares, it allows you to explore the restaurants and bars nearby. A short walk away from the sea organ and Sun Salutation, and really close to nightlife. Friendly staff. Only downside is wifi but it's vacation and time to unplug anyway!
Villa Skansi is a beautiful villa located on top of the hill with nice, modern loungey furniture and super friendly staff. They have a great party vibe going, which is definitely awesome if you're 25 and under as most people at the hostel are. I like to party just as much as the next person but felt much older than all people there (I'm 32). That aside, I moved rooms the second night and had different experiences with each gang so it's definitely worth the stay.
This hostel is located right at the edge of the Old Town/Diocletian's Palace area, steps away from some shops and plenty of restaurants and bars. The staff is really friendly and helpful, and the hostel itself has enough amenities to keep you busy should you want some chill time (although why would you want given the awesome city that awaits you outside?!). The riva is within a few minutes walk, and beaches are within 15-20 minutes walk. Great experience all around, I'd definitely go again!