you won't be a real stranger here. 70% of population here are foreign workers from all over the world. Everyone had the same concern of being along, as majority didn't have friends prior to coming here.
Qatar is safe country, there is nothing to worry about.
Women are semi-free, but it is unlikely that you will be asked: "Woman, where is your man? who do you belong to?". You can go around along or with other girls/guys, no problem.
Still, you have to dress modest, you are not allowed to live with man out of wedlock. Some companies just cannot accept a Lady-manager. Other than that, in daily life you will be fine.
Most likely you will be sharing accommodation with other nurses, so here you go, you will get friends in no time.
QAR 6800 with accommodation provided by company sounds not too bad. If you don't buy a car, your monthly expenses can fit QAR3000. With not too expensive car, add another about 1600-2000 a month.
If you don't have any job in your country, go for it. If you do have, think 10 times before leaving your home.