This is the same anon that asked this question and outside of my permanent capacity for procrastination towards doing things I should so I’m not sure why it’s taken me so long to answer it. The second part is already answered in the link. As the basis for my answer I’m reading Estranged Labour, one of Marx’s 1844 Economic and Philosophical Manuscripts available on the Marxists.org website which I’ve provided the link to.
So the question as I’ll be answering it essentially boils down to: how do we see the body in terms of it’s relations to private property?
The important thing to recognise in this is exactly what private property is. In the Marxist sense of using the term, private property is not simply something you own but instead something used as a tool of exploitation.
So there is a distinction between things you simply own (for example you may own a car, which would be your personal property) versus something you own but use for the exploitation of people. A capitalist owns a factory. People apply their labour in the factory to create products the capitalist sells for profit, and gives some of those profits to the workers. The capitalist’s ownership of the factory is what enables him to use the labour of the workers for his own benefit, which makes it private property.
This could be further deconstructed so for example I have a TV, that is my personal property. If my friends want to watch it I could charge them to do so (if I’m a seriously selfish person) at which point it becomes private property, because I’m using it to exploit others. Their money represents, at some point in the line, their labour which I am benefitting from for no reason other than the fact that I have something they want and I won’t give it to them for free.
So then how do we relate to private property, property which is used to exploit us? Because the workers do not possess the means of production, the profit from their labour is not theirs. Our labour is the physical action from our body. So the action of our body creates labour, and labour is applied to the means of production, which is the private property owned by capitalists.
The action of this alienates you from your labour. You work in the factory to make a product, the business owner sells the product for their own profit and gives some of that to the worker for their labour as a wage. The wage does not reflect the amount of labour applied to making the product. The wage comes from a variety of factors: the value placed on the use and exchange value of the product, but more importantly the capitalist will take as much profit from the product as they can whilst still retaining the workers labour.
In this way the labour is objectified. It is no longer an application of the body as a part of the self, a way of realising your work into the world, but instead this realisation is lost. You become alienated from the product of your labour because it’s purpose is for someone else’s profit. The product of the labour is an alien object. However the worker is bound into creating such alien objects, for the profit and enjoyment of another, because they become enslaved to the wage system. Without the stipend given by the owner, the worker cannot survive. The worker may change jobs but they cannot escape the system of alienated labour. (Well, short of becoming bourgeois themselves, but in doing so they become the exploiters rather than the exploited.)
Without this becoming a full blown essay I’m not sure how much more I can write on the subject. I really would recommend reading the manuscript I’ve linked to which goes into much more depth on the subject.
It has been suggested to me that there may be a third meaning in your question. The first potential interpretation I took to mean whether or not a person can sell their body in a metaphorical sense which I answered previously. The second I took to be regarding the way in which the body interacts with private property, the way that the application of the body through labour produces alienated products. The third potential meaning is on perhaps on some levels less personal and on some levels more personal sense. So relating to the medical field.
Firstly the idea of selling parts of your body (sperm/egg, blood or organ donation, selling hair to wig makers) which I don’t know if Marx ever wrote anything on. I don’t think it’d be viewed as anything specifically different. Your body is part of the self, selling it or aspects of it creates an alien object just as the product of labour but it isn’t private property in the sense that you can use it to exploit people. It isn’t the property of someone else, the state, the collective. It’s not even really the property of yourself, it is you. If that makes sense?
The other twist is, as reinventionoftheprintingpress, pointed out some big medical companies have patents on aspects of the human DNA which has been used exploit the nature of medical research for the benefit of the company with the patent (for example see this link). I don’t really think I need to spend much time explaining that this is an abhorrent practise. Such information, knowledge or research should be a part of the commons. Not least because of how disgusting it is to prevent medical advancement for profit, but because the idea of owning a part of everyone’s very genetic make up is completely bizarre.
Or was the question meant in some other way that I’m not realising?