Why am I getting this error? It says 'Local Player' (a nil value)

This isn't in a LocalScript. You can only access the local player from a Local Script.

Also, you cannot put this code in a local script as you index the ServerStorage and that is only accessible to the server, if FE is on.

Put this in a Script

local object = game.ServerStorage.Pistol
local CloneObject = object:Clone()

local function onTouch(hit)
    local player = nil
    for _,v in pairs(game.Players:GetPlayers())do
        if v and v.Character and v.Character:IsAncestorOf(hit)then
            player = v
            break
        end
    end
    if not player then return end
    CloneObject.Parent = player.Backpack
    print('hi')
end

script.Parent.Touched:Connect(onTouch)

EDIT: Fixed it there.

Scripts are global sided, and that's why there's no Local Player (except being in Roblox Studio)

You can try this:

local GUN_NAME = "Pistol"
local OBJECT_TO_CLONE = game.ServerStorage:findFirstChild(GUN_NAME)

function touch(part)
if game.Players:findFirstChild(part.Parent.Name) then
local player = game.Players:findFirstChild(part.Parent.Name)
if player:findFirstChild("Backpack") then
local CLONED_OBJECT = OBJECT_TO_CLONE:Clone()
CLONED_OBJECT.Parent = player.Backpack
end
end
end
script.Parent.Touched:Connect(touch)

You are trying to get player with LocalPlayer which can only be used on Client's Scripts.

Try this instead, and I cleaned up some code:

local object = game.ServerStorage.Pistol

local function onTouch(hit)
    local Player = game.Players:FindFirstChild(hit.Parent.Name)
    object:Clone().Parent = Player.Backpack
    print('hi')
end

script.Parent.Touched:connect(onTouch)