How deep should I sink a  gatepost for a 7' wide gate in my fence?
Rule of thumb is that 1/3 of the post should be in the ground. A 6' fence should be sunk into the ground 3', so you'd need 9' posts. 

As Shirlock states, even that won't likely support an 8' wide gate. Two 4' gates would definitely be a better option.

Perhaps the easiest solution, however, is to not even make a gate. Instead, build your fence and then make this an 8' panel that is attached with temporary fasteners...such as bolts. On the rare occasions that you need access, unbolt the fence panel and remove it completely--thereby negating the need for the two posts to cary the load of a cantilevered gate.

On the plus side, it's less gate hardware you need to buy and deal with.