There are no legal barriers to doing this: employee-owned companies do exist, but they're rare.
The average worker does not have that sort of money sitting around. Typically they want to own their homes, and if they do that nearly all of their money is going in that direction. They also want to save for retirement. One could argue that when they retire, they can sell their stake in the company and use that for retirement. I will show why that is a bad idea later.
Also, should such a company hire someone who is broke, like a young person entering the workforce? How would that be arranged? How would those people get jobs?
From a viewpoint of risk diversification, you are already over invested in your employer just by working there. If your company does badly, especially badly enough to lay you off or go out of business, the consequences for you are catastrophic. If all you own is stock in the company, your stock has depressed, maybe even zero, value at exactly the same time as you lose your income, making it difficult to survive long enough to get a new job, or move to a new job, or retrain for another profession.
Many companies offer stock options to their employees, and stock-purchase plans, where employees are given the chance to buy company stock at a discount as part of their compensation. It makes sense to do this, because it is in the interest of the company to have its employees' self-interest tied to the firm. The fact that it's a good thing for the company doesn't mean it's the best thing for the employees.
Many people feel that because they work at a company, they have an intimate knowledge of how well it is performing and how well it will do in the future. This is not accurate. Studies show that employees really don't have an accurate estimation of their companies. All you know firsthand is how well your part of the company is doing. Most of what you hear is one-sided propaganda from management, spun to put the best possible face on everything. It's common for layoffs to be preceded by an announcement to the effect of "We're facing a tough stretch ahead, but no matter what, we won't lay anybody off!". Furthermore, what you hear from your peers in the workplace is skewed. People know that expressing negative opinions about their employer is not a good career move, so you won't get an honest assessment from most of your co-workers. Furthermore, employees who do have insights that the company is going to do badly find other jobs and quit, at which point they are usually dismissed by the remaining employees as malcontents. So employees usually have a very distorted, rosy view of the prospects of their companies. Not a good position for an investor to be in.
Several times I have worked for companies where company stock has come into my possession through stock options or stock-purchase plans. I have always sold it immediately and invested it in something else, to diversify my position.
Similarly, using stock in your employer as a retirement investment is a terrible idea. What if the company goes bankrupt two years before you plan to retire? Hope you enjoy the taste of dog food!
When a normal company has a union, there is management independent of the union to represent the interests of the stockholders and customers, to balance out the voice of the union.
Would an employee-run company ever decide to buy new automation so that they can get more done with fewer employees? That is something that must be done periodically to keep the company efficient and competitive. If business is dwindling, would an employee-owned company lay off employees, or would they just sink deeper and deeper into debt until they go bankrupt altogether?