Actually it still won't work. For without the private and separate ownership of the means of production their can be no no economic calculation and no rational economic decision making. We can even assume their is no greed and envy and perfect communist person willing to give it his all to the community and the whole thing still couldn't function.
Meh. It's a simplification of what went wrong. I'm not a Communist myself (I'm a mutualist/pluralist anarchist willing to settle for radically democratic socialism), but I'll speak up in defense of the comrades.
The problem with Communist societies (stateless, democratic societies wherein the maxim 'from each according to their ability, to each according to their need') is in the method of revolution. You can go the Autonomist/Left-Communist/Council Communist/Anarcho-Communist route and have a mass worker's insurrection creating a communist society. Such societies tend to be short-lived, not because of internal instability, but because of an inability to defend themselves from external attack. However, internally, they tend to work out more or less as Marx wanted (though admittedly, we've yet to see how long this lasts on a large scale, because the large-scale ones tend to collapse from external invasion. However, many small-scale examples have been successful, and the Zapatistas have been around for over a decade now and indicate that the model can work well for at least that much time).
To solve the problem of mass rebellions being hard to orchestrate and of worker's republics (such as the Paris Commune, Anarchist Spain, Anarchist Ukraine, etc) falling, the International and the communists (after kicking the Anarchists out) settled on the idea of an intermediate socialist state to guide the revolution to Communism and defend it from reactionaries (capitalist states, aristocracy, nationalists, etc). Hence the 'dictatorship of the proletariat'. Note that the proletariat is a class, so a dictatorship of the proletariat is literally supposed to be the rule of the working class, through democratic worker's councils, or soviets.
The Bolsheviks in Russia took this a bit further thought up the idea of the Vanguard party, which basically states that a party of dedicated revolutionaries needs to lead the soviets to communism. This vanguard party was successful in warding off the many, many enemies of the revolution (including the White Army and invasion from Germany, Poland, the WWI Allies, Japan, and, later, the Nazis), but also took control from the soviet councils, ended democracy, instituted political repression, committed crimes against humanity, and killed off their political opponents, such as the Mensheviks and Anarchists. This all created a society that was not a federation of soviets, as originally intended, but a military dictatorship of the Party, which became a new class in itself. The means of production were not controlled by the workers, but by the hierarchy of the state.
No. There are bush tribes in Africa that at one time had absolutely no property. Nobody owned anything. Everyone shared what they had with the rest of the tribe. Native Americans didn't believe you could own land. Only share the land with the gods and the tribes.