"No Person shall be a Representative who shall not have attained to the Age of twenty five Years, and been seven Years a Citizen of the United States, and who shall not, when elected, be an Inhabitant of that State in which he shall be chosen."

"No Person shall be a Senator who shall not have attained to the Age of thirty Years, and been nine Years a Citizen of the United States, and who shall not, when elected, be an Inhabitant of that State for which he shall be chosen."

"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."

Besides, as soon as the President is sworn into office, he's the Commander In Chief of the armed forces, so if he wasn't already in the military he is at that point.

It was an unwritten requirement because nobody took non-veterans seriously as presidential candidates.

According to the table on that page, 11 US Presidents had no prior military service. This goes all the way back to John Adams, President #2 and our first Vice President. That's a full 25% that were not in the military. Many of the others were in the militia or national guard, which until fairly recently was not the same thing.

Military service was simply a huge political boost. Soldiers like Grant and Andrew "Stonewall" Jackson used the popularity generated by their military successes to launch political careers. During the 2nd half of the 1950s, the constant state of warfare (real or cold) generally meant the American public preferred candidates with military experience over those without. Once the Cold War ended, that preference died.