Hail of Stones
Type: Event Sphere: Tactics
Cost: 1
Action: Exhaust X characters to deal X damage to an enemy in the staging area.

ownership of characters is not defined in this card. I'm sure it should only be your characters but from the way it reads I could just pick anyones characters to exhaust to deal damage.

You can only exhaust characters you control to play Hail of Stones. From p. 25 of the rule book:

"Paying Costs
Many cards are written in a “pay or exhaust X to do Y” manner. When confronted with such a construct, everything before the word “to” is considered the cost, and everything after the word “to” is considered an effect.

Costs can only be payed [ sic ] with cards or resources that a player controls . If an effect is canceled, the cost is still considered to have been paid." [emphasis mine]