Usually when people talk about problems with Objectivism, they discuss theoretical and philosophical problems.
However, there is one giant practical problem with Objectivism that has probably done more than anything else to prohibit an Objectivist society from actually existing during as long as any of us have been alive.
People with very strong group identities who think in terms of group rights (not individual rights) and may even have very strong group grievances always end up “hijacking” objectivist societies away from the atomized Objectivist individuals who comprise said societies (and I’m not just talking racial groups here, I’m talking about class groups, etc).
The USA is a case-in-point. Ayn Rand often lauded the USA as it existed during the period of roughly 1776 until roughly the beginning of the 20th century as the ‘one bright spot of history’ that existed between a ‘dark period’ before the USA’s founding and another ‘dark period’ that supposedly began sometime in the early 20th century and that continues unto this day. In our grandfathers’ day (the early 20th century) it was (here in the USA) in the form of a “class conscious proletariat” who banded together to usher in new things like the New Deal overtop of the pleas of the individualists. In our day, it’s racial groups. For instance, it’s the so-called “crybully movement” who are calling for special treatment for various racial groups on the grounds of past grivences such as slavery. And they’ve been remarkably successful. Libertarians and objectivists have been plowed under, frantically waving their hands around begging for “individual rights” while crybully grievence groups laugh in their face, truly knowing that they have the power.
Now, I’m going to say right off the bat that I don’t necessarily think that being someone with a very strong group identity is necessarily worse than being an atomized libertarian individual. In fact I think it is better (although I do think it is wrong to use that for the purpose of crybullying).
In fact, I would even go so far to say that the kind of society that Objectivists plead for eventually necessarily leads to socialism and communism. Just look at America, just look at all the places that have suffered through Communism. Are those places better off or worse off than before the Enlightenment revolutions? Was Russia better off or worse off than after the February 1917 revoluton which toppled the Tsar (which Rand was for, by the way)?