Healthcare (beyond eating well and exercising) is not something that people can provide for themselves. Try as you might, there are not enough triple shifts in the world to prevent cancer or heal a broken leg. Citizens should not have to pay to be healthy, especially for the many countries that promise them security, shelter, and freedom.
Yes, healthcare should be the responsibility of the federal government. The United States is virtually the only country in the developed world to not offer free healthcare to all of its citizens. As the world's richest country, the United States should easily be able to afford to provide healthcare to all Americans.
Everyone should be able to access good quality healthcare and in a capitalist system this is not possible without the intervention of the federal government. Healthcare should be a right and the government is best equipped to make sure that it is affordable and effective for everyone. Private companies interested in profits will never be able to provide affordable healthcare for everyone.
If affordable healthcare can be offered to the public through employers and some type of coverage is offered for retirees and the unemployed, then the government shouldn't be responsible. However, the federal government must offer some type of health care for retirees or the unemployed if there is not other solution.