The British Land Pattern Musket, nicknamed the Brown Bess, was a flintlock smoothbore that entered service in 1722. It didn't leave general service in the British military until 1838. Worldwide, its service lasted for at least 60 years more. There are accounts of it at the Battle of Shiloh in the American Civil War (1862), and I have accounts of its use against U.S. soldiers during the Bombardment of Ḵaachx̱an.áakʼw in 1869.

Colt .45 m1911 pistol is still used by some military operators.

Browning 50 caliber machine gun is coming up on 100 years as well.

The Douglas DC-3 is still in use by small operations in both passenger and cargo service; its first flight was in 1935. The military version, the C-47 Skytrain, is still in (very limited) active duty with the Greek Air Force, from certain reports. Colombia is using a AC-47 "Spooky" gunship variant, as well as a few other nations.

The Antonov An-2 "Colt" biplane was first flown in 1947, and is still in production in China as the Y-5.

For a weapon still in service in some military and that is more complex than just a metallic blade, I would put my money on the Mausin-Nagant Rifle from 1891. Nearly 125 years of active duty, still used by the Finns and ex USSR countries.

For an artillery system : Close to a century is the Gribeauval system. 80 years of active duty.

For a mbt, the T-54/55 by far. It is the direct sucessor of the famous T-34. That's the busted tank you often see in middle eastern war videos (Along with it's T62 sucessor). A variant of it it's used by the israelian army as an apc. For having seen it in real life, it's main limitation is the size of the habitacle : if you are higher than 175cm, you will have a hard time fitting inside. It already have nearly 70 years of active duty and will most likely be still here for an another 20 years at least.

For a plane, The Antonov 2 and the DC3 are more or less tied. But the An-2 is still produced and seems to have many years ahead of him. Both have more than 70 years of service, with the an 2 going most certainly for the 100+ years.

The standard German military light machine gun, MG3, is pretty much a slightly modified MG42 and I guess it will stay in use for some more years to come.

The Carl Gustaf Recoilless Rifle (GRG m/48) was developed in 1948 and is actually seeing increased use (in a slightly updated version) and the US in 2011 ordering the weapon for standard infantry units (earlier only being used by special forces in the US).

Walls of Constantinople stood for more than 1100 years before being breached. Then repaired and useful for some time after that.

I'm not sure if you mean to ask only about modern-era machinery and ordnance, but there are quite a few examples of military hardware that saw centuries-long use in pre-modern history.

Variations of the gladius hispaniensis were used by the Roman armed forces for about 500 years, from the 4th Century BC to the end of the 3rd Century AD, and even probably beyond that in less widespread capacities.

If you mean more modern technology, the Minuteman series of Intercontinental Ballistic Missiles was first deployed in 1961 (the current variant was deployed in the early 1970's), and the Air Force plans to operate them until at least 2030, probably beyond. That's not quite 100 years, but time will tell.

A big category that would get you lots of responses would be ammunition. The Browning .50 caliber bullet was first developed around 1910 for use in a heavy machine gun, and still sees service in such guns today, as well as high-grade anti-materiel sniper rifles. Other bullets, like the 9mm Parabellum and .45 caliber bullet are nearly as old and still very popular.

And then it also depends what you mean by "continuous use." Do you mean continuous use by the same military? If so, you'd narrow it down a bit. But if you mean continuous use by any military, then there are a lot that come to mind. The Russian T-54/55 main battle tank, for example, was first developed in 1946 and continues to be used today, in several variants, by countries around the world, and will likely continue to be used for at least another 50 years.

The Maxim machine Gun from the 1900's is still used by some russian troops as well as Eastern-Ukrainian militias in the ongoing Donbass War

Sorry, we don't allow throughout history questions. These tend to produce threads which are collections of trivia, not the in-depth discussions about a particular topic we're looking for. If you have a specific question about a historical event or period or person, please feel free to re-compose your question and submit it again. Alternatively, questions of this type can be directed to more appropriate subreddits, such as /r/history or /r/askhistory.