Since The Constitution is the limiting document upon the government, the government cannot become greater than the granting power. That is, the servant cannot become greater than its master. Therefore, should the chief executive or the other branch of government or all branches together act to suspend The Constitution under a rule of martial law, all power granted to government would be cancelled and differed back to the granting power. That is the people. And I'll conclude with this statement: Martial law shall NOT be possible in this country as long as the people recognize the bill of rights as inalienable.
The problem is that the Constitution specifically allows for the imposition of martial law, suspension of habeas corpus, in certain circumstances.
Article 1, Section 9 of the U.S. Constitution states, "The Privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it."
The Article of the Constitution deals with the powers of Congress, so there can be no doubt at all that Congress has the power under the Constitution to suspend habeas corpus when necessary. There might be disagreement over whether a particular suspension meets the criteria given, of course.
AFAIK, there has only been a single instance of suspension by the president. A. Lincoln during a period when Congress was out of session and there can be no doubt at all that Rebellion and/or Invasion was actually under way. Even in that case, Lincoln applied to Congress when it assembled to retroactively approve his suspension, which it did. All later Civil War suspensions were passed by Congress, not be presidential proclamation.
And Congress could suspend “habeas corpus” under certain circumstances. The articles do not include suspending the Constitution, as if it is suspended, then as Norman pointed out, the Government is no longer in existence.