Q:

Is solving unemployment impossible? 

Supposing that there in island.  The island is barren except for a machine that makes food.  Let us suppose that people want for nothing else. 
At first the machine requires the whole population to maintain it and each is paid with food tokens. 
But gradually the machine is updated until it requires only one person to run it.  This person is paid in food tokens.  He can use some of the tokens to buy security to stop other people stealing his food. 
Now it seems if the market is allowed to run its course, this will end up with half the country with no food and no jobs. 
If nature takes its course, this half of the country will either die in a famine or simply decide they are too poor to support children and the population will drop. 
But then the person who owns the machine needs less security, and so sacks them.  Eventully it will just end up with him by himself. 
The alternative would be to enforce giving free food to people without jobs. 
Is there an economic solution to this in which the population is maintained but everyone is not on welfare?  Basically giving everyone the same even though one person is doing the work. 
Or are these the only alternatives:

Survival of the fitness, let the population die off
Give everyone free money even if they are not working. 

These seem kind of like the left/right political spectrum but neither seems ideal. 
(BTW.  As I said the humans on this island are totally satisfied only by the food.  And so there is no "solution" which involves things like becoming a masseuse or a poet or a comedian.  One can assume that there could equally be a machine that could do these things.) 

A:

Let me rephrase the question in my own words: in an environment where automation increasingly is taking over, are there alternatives besides leaving the have-nots behind or providing a basic income? 
An (non-ideal) alternative is to pay people to do a job that is pointless (not related to food production in your analogy).  If the point of this question is to confirm your belief that there is no other ideal alternative within your limited analogy, then you have a +1 from me.  
I would also argue that basic income isn't necessarily politically left wing, but I'll just link you to Politics SE to debate that.  And perhaps I am too left wing to see otherwise but if there is a magical machine as in your analogy that takes care of food for everyone, why not just distribute food unconditionally?  How is that not ideal?  I understand the argument that humans need to work to fulfill "higher needs" but I think that is a separate question from needing to work to fulfill physiological needs.