Vali's is the closest you'll get to a backpacking hostel in Mashhad. He offers a twin-bedded guestroom in his house, along with a number of sleeping cots in a large communal room and on his terrace. You can also get great, home-cooked meals here for a decent price.
If you'd like to book here, you can find booking and contact information here on Lonely Planet.
Valis Non Smoking Homestay, Mashhad, Khorasan Razavi, Iran (Directions)