The Marxist critique of private property and capitalist wage-labour appeals to the notions of exploitation and alienation, though they miss an alternative objection which I find much more compelling.
'Exploitation', in the technical Marxist sense, is when workers are not fully compensated for the value of their labour. The capitalist appropriates profit for himself, and so is said to 'exploit' the workers to whom this 'surplus value' properly belongs. But this is not necessarily unjust, especially if the workers have consented to volunteer their labour in such a way. And even if forced (e.g. due to a lack of acceptable alternatives), such transfer of surplus value may be justified if, say, part of a temporary apprenticeship that everyone must pass through. So such 'exploitation' is not intrinsically wrong. (Further, it rests on the faulty libertarian notion of absolute property rights, of which we should be skeptical. We do not want to say that redistribution to the needy 'exploits' workers in the Marxist sense!) The real injustice must rest elsewhere -- perhaps the lack of equal access to the means of production, or the power imbalances, or some other egalitarian concern. As such, it might require the redistribution of private property, but not its outright abolition.
Marxists also argue that the capitalist system 'alienates' workers from their essential human powers. They suggest that our 'species nature' (cf. Aristotle's "rational animal") is to engage in freely co-operative creative production. Much modern labour (think factory-line mass production) is a repugnant perversion of this ideal. There's no creativity involved, no craftsmanship or intimate connection with the produced object; indeed, many workers may not even see the finished product. Most importantly, wage-labour commodifies our labour-power and places it under others' control. Marxists see it as an absolute wrong to alienate us from our essential natures in such a way -- a wrong that will arise from private ownership of the means of production.
There are two main problems with this view. Firstly, the focus on 'species nature' strikes me as misplaced. As an account of human nature, it is descriptively false and ethically irrelevant. The focus should instead be on recognizing the important contribution that satisfying work can make to human well-being. Secondly - and relatedly - it ignores all competing goods. Perhaps we would be willing to suffer through some unrewarding and 'alienating' work in return for the wealth to buy luxury goods. At the very least, this possibility should not be ruled out in advance. The Marxist is unreasonable to suggest that unalienated labour is the supreme value in life!
Nevertheless, I do think there is an aspect of the 'alienation' argument that's quite compelling, and which implies that wage-labour constitutes a form of moral degradation, namely, prostitution. Just as the sex-worker prostitutes her body for economic gain, so does the worker prostitute his labour. We submit our person to the power of bosses, who can then exert almost arbitrary control over our working lives. This is morally degrading. The worker is objectified, and used merely as a means for the employer's benefit. In a more just system, people would not have to give up their humanity for forty hours (or more) a week.
Perhaps it's unavoidable, I'm not sure. At least there seems to be room for improvement -- and I would like to think that we could develop an economic system whereby socially necessary goods could be produced in a manner consistent with human dignity. I suspect most people would be willing to sacrifice some degree of material prosperity to achieve this social good. But of course the criticial question is, "how much?"