Cut across the trunk of the tree approximately 3 feet above ground, using a chainsaw to chop any remaining portions down. Remove any trunk or branches from the ground to clear the area.
Dig a circle around the outer perimeter of your oak, working 3 feet away from the base. Dig downward to a depth of 12 to 18 inches, removing dirt as you go, to clear the hole and expose the roots.
Use a pickaxe, or the larger Dutchman's hoe if desired, to break apart dense clumps of soil and work around rocks. Work carefully not to cut through major roots 2 inches in diameter or larger.
Continue to dig until you have exposed the majority of the oak's root system immediately below the trunk and inside your 3-foot perimeter. Trace the larger roots away from the base of the tree, using your shovel or pickaxe to expose them beyond the perimeter and loosen them from the ground.
Grip the base of the trunk and shake it back and forth to make sure it is loose in the ground. Expose any roots which prevent the tree from shaking. Continue until the tree is resting in the hole, but is no longer attached to the ground.
Raise the trunk from the hole by hand or by wedging the shovel under the base and using the shovel as a lever. Fill in the hole with the loosened soil.