Is it possible to get a pet in Skyrim?

Yes. You can summon one for a short while, or summon four "different" permanent elementals, ask one to tag along, or buy one.

http://elderscrolls.wikia.com/wiki/Meeko%27s_Shack

You can get this dog just by simply going to near his house. Make sure you follow him when you first see him. Afterwards, unlike the dogs you buy, you can tell him to head home and later pick him back up at the shack for free again. He does get annoying some times, since he'll run blindly into traps and bark over and over AND OVER AND OVER during important events.

If you want a more fixed way to get a pet, you should be able to buy a dog just outside of Markarth.