That is utterly false. Even the Second Amendment text itself requires those who keep and bear arms to be "well-regulated," and the Fifth requires the protection against unnecessary loss of "life, liberty and property," which unrestricted firearms certainly threatens.
They may not be able to ban firearms, but they can indeed restrict them to a limited extent.
The well regulated are the Militia (the army). not the People, whose right to keep and bear arms shall not be infringed.
This came up in the debate over whether there should be a Federal army at all (see: The Federalist Papers).
One, of course was necessary to settle (if need be) disputes between the armies (militias) of the several states, and to prevent invasion of the states of the federation. However, the Founders saw it necessary to prevent the Army from taking over, as they had just overthrown a government imposed by military governors on many of the colonies.
By what means would the Army be kept under civil control? By the force of arms, if necessary, of the people, who even lacking martial training would overwhelm any effort by the military to impose tyranny by sheer force of numbers, armed with their personal weapons.
Now, let's read the Second Amendment again:
A well-regulated Militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed.
It is the people who regulate (control) the Militia (Army), by virtue of being armed.
The only infringement on the Right to Keep and Bear arms was one of what you could afford.
Similarly, the restrictions of the Fifth Amendment are restrictions placed on the legitimate abilities and powers of Government. The people were already restrained by criminal laws against murder, stealing, and such, but the government could theoretically pass a law which made it okay to steal from the people (an argument against "civil asset forfeiture", which, imho, is a clear violation of the Fifth (and Fourth) Amendments or round them up and imprison or kill them.
The Bill of Rights isn't there to impose restrictions on the Rights of the People, but to ensure those Rights are protected from the Government.