Arbitrary in the sense of humans judge the entire thing. A human places the ball (prone to error), humans move and place the yard markers (prone to error), when they measure for first down they place the one end (prone to error), then they pull the other one to the end of the chain (tell me when you see them pulling out the slack, etc that isn't prone to error) to then eyeball whether the tip of the ball is past the marker. Now, the theory is that in the end it all balances out one way or another in terms of luck of placement or that the cumulative error-proneness often ends up making the end result more approximately correct than some of its components may be. Nonetheless, add that all up and ultimately all of the placements, ultimately all subject to human error (lets not even get into dudes throwing their hats away from their body and the like to approximate where something landed) and thus in that way arbitrary.
Technically, with things as they are, they could place something in the ball itself and you could begin to make things painfully precise... or take major steps towards it. Would that make the game better? No. But if accuracy was a concern they would do something about it.