Well now, only if your ready for the responsibility. And not a real dog mind you, but if you do what the dude below me says then yes you can get a real pixel Dog!! Woot Woof. pun intended.
If you take the dog with you in the Human Noble origin, or if you find a "Wilds Flower" for the Kennel Master in Ostagar so he can cure the dog, who'll join you later on when you leave Flemeth's hut.