While the Founders believed that a standing army would too easily be a tool for tyrants, and wrote the 2nd Amendment to negate this possibility as much as possible, they did recognize the need for a standing NAVY.
Remember the tech of the times. In the 1780s, sailing ships were the most complex and powerful weapon systems that existed. You couldn't call up farmers and train them to run a ship in a couple weeks. You needed permanent crews.
Navy's had three primary functions, deal with other navies, exercise power against shore (to the range of their cannon) and most importantly, rapidly transport troops and supplies over long distances.
Doing that well meant you needed trained and experienced people, and the only way they get that way is on the job training, so a standing Navy was a necessity, where a standing army wasn't seen to be that way, since militia's could do the Army's job, mostly.
The Constitution doesn't expressly forbid a standing army, so, saying a standing army is unconstitutional is not correct.