From a Constitutional perspective, it is a strange term, because the Constitution is (somewhat) clear about what the requirements are to be President. That is, the rulebook lists the three rules that must be followed, and if any of them are violated, that makes that particular individual no longer qualified to be President of the United States. Found in Article II, Section 1, Clause 5 (and as a side note, you can go to your local Congressperson's or Senator's office and request a free copy of the Constitution!), “No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty five years, and been fourteen Years a resident within the United States.”
1. No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President - This criterion offers two avenues to citizenship. The latter expired at the signing of the Constitution, and the former has been discussed at great length, beginning with President Obama, but also in the context of Sen. Cruz in the current election cycle, and in both cases, revolving around the birthplace and citizenship of each individual. It is undisputed that both Sec. Clinton and Mr. Trump are citizens of the United States.
2. No person shall be eligible to [the office of President] who shall not have attained to the age of thirty-five years - I examined over two-hundred years of Supreme Court case law in researching this point, but there has not been one case involving a challenge to the President on the basis of not having the requisite age. The youngest President was Theodore Roosevelt (26th President, 43 years old), but there has been no challenge to this provision, and at 69 and 70 years old, respectively, Sec. Clinton and Mr. Trump are both qualified in this regard.
3. No person shall be eligible to [the office of President] who shall not have been fourteen Years a resident within the United States - There is some evidence that the Framers believed the fourteen-year residency requirement could be satisfied cumulatively, rather than consecutively. An earlier version of the clause excluded individuals who have "not been in the whole, at least fourteen years a resident within the U.S." (emphasis added), and historical evidence suggests that deletion of the phrase "in the whole" was not intended to alter the provision's meaning. President Hoover, whose successful 1928 campaign won him the Presidency, came less than fourteen years after his return to the United States from foreign residency in 1917. Relevant to the modern day, there may be unsettled law about whether an individual who serves in the military or State Department and resides overseas would be eligible for the Presidency due to this clause. However, in this election, both Sec. Clinton and Mr. Trump appear to meet this criterion.
On its face, at least, both candidates appear to meet every legal requirement for the Presidency. Glaring by their absence from this list are extra-legal characteristics that are commonly attributed to be disqualifying, including, bit not limited to not having charisma, or having a scandal or being ill-tempered. However, there exist other legal barriers in addition to that which is listed in the Constitution. Relevant to Sec. Clinton, 18 U.S.C. 2071 is sometimes cited as a statute which may disqualify her from service. Titled "Concealment, removal, or mutilation generally," the statute states that "Whoever, having the custody of any such record, proceeding, map, book, document, paper, or other thing, willfully and unlawfully conceals, removes, mutilates, obliterates, falsifies, or destroys the same [...] shall forfeit his office and be disqualified from holding any office under the United States." To be clear, this is a federal criminal statute that would require a prosecution under it, and a conviction by it, to even begin to be relevant, but to the notion of creating additional requirements to discern a President, it is relevant now.
In a wholly separate series of cases involving term limits, the states, and specifically, Arkansas, tried to add requirements on top of those set forth in the Constitution. In essence, Amendment 73 to the Arkansas Constitution prohibited anyone from appearing on a ballot who had already served two terms in the U.S. Senate or three terms in the U.S. House of Representatives. A candidate argued that the amendment violated Article 1, Section 5 of the Constitution, arguing that the Constitution establishes sole qualifications for federal office and they can not be altered by the states. Faced with the question of whether the states may alter the requirements set out in the Constitution for Members of Congress, the Supreme Court, by a vote of 5-4, the Court concluded that the Framers intended the Constitution to be the exclusive source of qualifications for Members of Congress. While this case dealt with the states, and the Court, to some degree discussed the Tenth Amendment, the ratio decendi of the opinion was this notion that the Constitution is the full and complete list of requirements to run for office.
Translating that analysis to 18 U.S.C. 2071, a similar argument can be made that would hold the statute invalid as creating additional burdens and requirements on top of what the Constitution requires. The three requirements in the constitution are negative requirements - "who shall not have reached..." - and it could be argued that the Framers could have contemplated excluding persons with convictions from running as a fourth negative requirement, but that the exclusion of that requirement and the inclusion of the other three provides evidence that the statute may be an unconstitutional addition to the constitutional requirements.
Is one candidate or the other Ill-qualified for the Presidency? Perhaps, depending on your perspective and belief, but likely not "disqualified." For why such a distinction matters, go to the second part of this post here.