What if you're unskilled and you work in a peach factory or something (for the sake of the argument)? Without a minimum wage, you could say that the market mechanism will work things out. Obviously nobody will work for free. However, if you're suited to a job in peach canning, you probably aren't very qualified (this could be your own fault, or it could be due to other unavoidable problems), and you certainly aren't in a very good bargaining position. If you run a household you will need some source of income (assuming that this economy has limited/no welfare scheme). So, your employee essentially has the power to determine your working conditions. You essentially agree to them out of obligation to your family. And still, what you're paid is barely enough to get by, and if an emergency happens you're put in a very precarious condition indeed. There is also a risk that there will be a lack of investment in your children's education, which may end up leading to future generations having to live the same standard of living as well. At least with the minimum wage, there isn't a threat that your wage could go any lower, assuming the company still needs the same amount of workers regardless of what they are being paid, and there is enough to give younger generations of unskilled workers a chance.
Ordinarily, I think that the market mechanism is essentially quite good at allocating prices and resources. I don't agree with supporting price fixing or uncompetitive industries. But I think labour is different to other resources because even though both the employer and the employee technically "need" each other in order to work, oftentimes the employee is in a position where they don't have equal power as the employer. When it comes to consumers and producers, there will usually be an "equality" of sorts (unless the good is technically a necessity such as education or health - which is probably why these things are greatly subsidised by the government), but that's not always true when it comes to labour, especially unskilled labour.
When I think about no social safety nets such as the minimum wage and workers rights, I usually think of England during the Industrial Revolution. The working poor, the unsafe conditions, huge divides between rich and the poor, little opportunities to "lower classes". Some governemnt intervention is good, it smooths business cycle fluctuations (Keynesian policies technically helped to get us out of the depression), it doesn't necessarily bring equality (which is good), but it does at least give everyone a certain amount of opportunity. Obviously if it's excessive, it's bad, for example, I think centralised wage fixing is a horrible idea, but again, there's a balance. Both extremes are flawed, I believe.