A extremely lazy girl that goes to the barber I remember watching a anime not to long ago. I don't remember anything but about what happed in the first episode. So I hope that is enough info. 

What I remember of the first episode is it starting of with a girl laying in bed with extremely long hair. And I believe her brother came to the room to tell her not to go out of her room as he had just cleaned. 

She decided to go outside for once. She was walking in the town looking for a barber, and she met some random guy here which takes her to a park and cuts her hair there. After which she went home without paying. After telling this to her brother he got pretty mad, saying she shouldn't trust people so easily and that she should go back to pay him for the hair cut. 

and I believe her brother said something that pretty much stated you are mine, and mine alone in the end of the first episode. 

Hope this is enough been looking around allot to find it again :)
By googling anime hair cut in park I got this:

Kuro to Kin no Hirakanai Kagi

The synopsis seems to match quite well with your description:


  Katagiri Kanade is a hikikomori who spent her days lazing off in her room. Her specialty is her broad imagination which often leads to various wild delusions. One night a weird cowboy guy appeared in her dream, saying that he's her heart's hero. Due to his encouragement, Kanade decided to try and go have a haircut the next day. It was then when she met a mysterious guy who cut her hair in the park, giving her courage to resume her school days.