What type of companies are completely dependent on the government?

Like the companies that build our roads, schools, and any other type of civil engineering. The companies that sell foods to schools for lunches, or sell needles for public hospitals.
These Companies have a direct monopoly from the government because they are guaranteed business.
My questions is.

Are these businesses privately owned?
How did they get the backing of the government
and how do i get in?

Most companies who perform such services as the above listed, are privately owned. They are considered in the "Private Sector" but do work almost exclusively for the government. While being privately owned, the government may also choose to subsidize or give money to them for research etc.

Getting a job in one of these companies is not much different from any other company, but depending on how close the company is dealing with national security, the government will probably run background checks. If you have been convicted of a crime, you may have less of a chance of getting the job because of its relationship with the government.