I was originally thinking along those same lines. However, a base quest isn't one you receive from your fellow survivors the base quest is the quest that pops up when you get near a base and it tells you that all zombies must die.
You can keep trying this one after game completion though it is more difficult because of the different types of zombies.
I did mine fairly late in the game at the police station base. To activate the quest simply approach the door and follow the button prompt.
If you get hit simply exit the game and reload. You should find yourself on that same map and go to the door to start the quest.
Don't try melee at all its just too easy to take a hit. I used the Level 2 sonic shotgun and the pistol/revolver.
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