Okay, others have given politically-slanted answers, so I'll give mine. Disclaimer: None of this is provable fact. This is my opinion of how it would likely work in real life.
The economy booms. Without government regulation forcing people to waste time with red tape and follow "one size fits all" regulations, industry is free to spend their time producing useful products and services.
Technology booms. Companies no longer need the government's permission to introduce a new product. For example, new medicines can be brought to the market very quickly, rather than having to go through years of paperwork and bureaucracy to get government approval. I understand in the U.S. it now takes the FDA an average of 12 years to approve a new drug. 12 years during which people die even though the drug that could save their lives is sitting on the shelf.
Bureaucrats do not rule people's lives.
There is great exercise of freedom. Many competing religions and philosophies are practiced. There is vigorous debate on all sorts of questions, from the most practical to the most esoteric.
Many of the political conflicts we see in our society go away if we adopt the libertarian concept that "rights" mean things you are allowed to do whether your neighbors like it or not, rather than things you can force your neighbors to participate in whether they like it or not. For example, consider the debate over gay marriage: In a pure libertarian society, a person who disapproves of gay marriage cannot prevent two homosexuals from declaring themselves married. At the same time, a homosexual couple cannot force someone to provide services for their wedding, like providing a venue or baking a cake. (I'm sure people on both sides will protest that this is outrageous. Those who oppose gay marriage might say that society should not condone this and homosexuals should go to jail. People who support gay marriage might say that failing to provide services to a homosexual marriage is discrimination and people who do this should go to jail.)
Problems that have obvious solutions:
Consumers will be victimized by unregulated products that are of low quality or even dangerous. People rely on recommendations from friends and neighbors. Web sites or other forums spring up where people can review products. Organizations are formed that test products and certify them for quality and safety -- like Underwriters Laboratories and Consumer Reports in the United States.
Big corporations will exploit the workers. No, they won't. If the company treats you badly, you quit and go work somewhere else. Or you start your own business. In a libertarian society, starting a new business is easy: by definition, there's no red tape. You just declare yourself to be in business one day and you are. Sure, finding a new job isn't always easy. But people do it all the time.
Big corporations will drive all the little companies out of business. How? In a socialist society, big companies can destroy little ones by having the government make all sorts of complicated rules. At the least, for the big company to hire lawyers to figure out the rules is a small expense to them, one or two more employees on top of thousands, while to the little company it's one or two more employees on top of the one or two they already have, a huge expense. At best, the big company hires lobbyists who get all sorts of exemptions and special cases put into the rules that benefit them.
Without government welfare programs, the poor will starve. Before there were government welfare programs, private charities took care of the poor. Yes, the system was overwhelmed to some extent during the Great Depression, which led to support for government programs. But it's not at all clear that the government programs worked better than private charity. Government programs are inherently bureaucratic and wasteful. Private charities can provide help on a case by case basis, helping the truly needy while rejecting people who just don't feel like working and want a free ride. If you don't like the way a private charity makes its decisions -- if you think it's rejecting people because of their race or some other unfair criteria -- then don't give them money. Government programs pretty much have to have a book of rules, which means people have an incentive to search for loopholes rather than just get a job. If you don't like the government's rules, you can't just not pay your taxes.
Problems that are hard to solve:
Private police forces protect the rich but not the poor. If private police forces can arrest someone, what stops the rich man from having his police arrest anyone he doesn't like, or who inconveniences him? What happens when two rich men clash in their idea of justice, whether a genuine philosophical disagreement or a conflict of personal advantage? Do their policemen fight each other? Personally, I'd say that private police forces are carrying libertarianism too far. I'd keep the police as one of the few functions of government.
Private roads If roads are all private, who maintains them, and how is this paid for? If a private company runs the road in front of your house, do they charge you a toll to use it? If you refuse to pay the toll, can they block you from leaving your house? If so, what stops them from charging outrageus prices? Unless there are two roads to every house, there's no competition. It's pretty tough to just do without, as that means not leaving your home. The only limit is when they bankrupt you. In general, if there's no "public property", then almost everyone's home is surrounded by other people's property. What if they all put up fences and refuse to let you pass? It seems a libertarian society has to have some deviation from strict libertarianism to solve this problem, some provision for public property, or a right to pass through other people's private property without their consent.
Pollution? Pollution can impose significant costs on people who are not parties to a transaction. i.e. if a factory produces and sells a product, and generates pollution in the process, people can suffer who are neither the seller nor the buyer, and so who in a purely libertarian society have no control over the transaction. It's difficult to see how this problem can be solved without some sort of laws regulating pollution.