This question reflects a deep and pervasive misunderstanding of Marxist philosophy, a misunderstanding that I suspect was intentionally created (somewhere in the early to mid 20th century) as a straw-man argument in the 'fight' against socialism.
When Marx talks about the abolition of private property he is talking specifically and exclusively about the private ownership of collective means of production. Marx is not particularly interested in the ownership of non-productive property: housing, food, clothing, etc. Such things only come into consideration, for Marx, to the extent that the private ownership of collective means of production becomes a system to deprive laborers of such niceties. Marx also doesn't worry much about an individual owning his own means of production. A carpenter can own a hammer — a tool he uses to ply his trade — and all is well with the world.
Marx is concerned about a specific feature of capitalist society. Let's say (extending the above example) that instead of every carpenter in the world owning his own hammer, six people with a lot of money buy up every single hammer on the face of the earth, and then turn around to rent those hammers back to carpenters. These six people — let's call them 'hammer capitalists' — are now in a position where they can dictate whether any given carpenter works, because a carpenter can only work by renting a hammer from one of these six people. That means these six are in a position to dictate whether a carpenter has decent housing, has new clothes, or even whether a carpenter has anything to eat, because a carpenter cannot do or have any of those things unless he works, and he only works if he rents a hammer from these 'hammer capitalists'. The 'hammer capitalists' can thus set their hammer rental rate arbitrarily high, forcing carpenters to work increasingly more merely to pay off the rent. The end result is that most of the carpenter's excessive labor produces value and income for the 'hammer capitalist', leaving the carpenter only enough to get by, even after long, long hours of work.
Of course, in the practical context of Marx's world we don't have 'hammer capitalists' who own all the hammers and rent them to carpenters. We have industrial capitalists who own industries, and rent 'employment'. That is the nature of an employment contract: an employee is given the right to use the capitalist's industrial property to produce goods, at the capitalists's pleasure; the capitalist takes all of the goods that are thus produced as his rent for that right, and offers the employee a comparatively small wage which the employee is obliged to take, because if the employee does not agree to the terms, he cannot work at all, and will not be able to afford the basic requirements of continued life. In its most degraded form this system is akin to slavery, where laborers labor incessantly merely to keep body and soul together, while owners sit back and reap massive profits.
The core of Marxist thought, setting aside all of its complexities for the moment, is the question of how workers can break free of this noxious 'rent' condition, gaining some ownership and control over the means of production they need to do work, so that their labor cannot be exploited and they can reap a reasonable profit from the products of their own labor. If you've ever wondered why lower income people often spend more than half of their total income merely to have a place to live, you've touched on the edges of Marxist thought. No one has yet come up with a good solution — and plenty of people have come up with bad solutions — but the question is a vital and important one. However, we have to recognize that the point is not the end private rights in some vacuous and draconian fashion. The point is that property rights, construed poorly and prejudicially, can become a tool in its own right for depriving other people of their own property rights. For Marx, a man who invests labor to produce a product has property rights over the product produced, regardless of who owns the tool used. He can sign those rights away — sell his product — if he so chooses, but he should not be forced to sign those rights away merely to meet the basic necessities of life.