The World's Shortest and Longest Wars

The Harem and Palace of Zanzibar were destroyed in the Anglo-Zanzibar war of 1896. The Palace was never rebuilt, but the Harem (of course) was later rebuilt.

The World's Shortest War

The Anglo-Zanzibar war, also known as the world's shortest war, was over in under 45 minutes. The exact length of time is actually debatable - some had put it as short as 38 minutes - anyways, it was a very, very short war.

The whole thing started when the Sultan of Zanzibar, who had willingly cooperated with the British, died on August 25, 1896, and his nephew Khalid bin Bargash seized power in a coup. Thinking that another candidate would be easier to deal with, the British delivered an ultimatum to force the Bargash to abdicate.

Bargash refused the ultimatum and assembled a navy in form of the ex-Sultan's yacht, the HHS Glasgow and fortified the palace. The British, on the other hand, assembled 5 modern warships in the harbor in front of the palace and landed two battalions of army. Bargash tried a last-ditch negotiation effort through the US Embassy, but time soon ran out.

At 9:02 AM on August 27, 1896, when the ultimatum ran out, the British navy sank the Sultan's fleet (yes, that one yacht), shelled and destroyed the palace completely. About 500 people, mostly Bargash's soliders, died. Bargash ran and hid at the German Embassy, where he was later granted asylum. Depending on who you ask, the world's shortest war was over at 9:40 AM or so.

The World's Longest War

The world's longest war, on the other hand, spanned 355 335 years without a single shot fired. That war, between the United Provinces of Netherlands and the Isles of Scilly, got started in 1651 and the story goes as follows:

During the English Civil War (1642 - 1651), the Parliamentarians beat the Royalists further and further away from London, until it was forced to retreat to the Isles of Scilly off the Cornish coast. The Netherlands, which sided with the Parliamentarians, sent the Dutch Navy to fight the Royalist fleet.

The Dutch Navy was so badly beaten that the Netherlands decided to declare war. However, they couldn't blame England, since it was the Royalists in Scilly that caused them so much problem - so they declared war on the Isles in 1651 instead. Later that year, the Isles of Scilly fell to the Parliamentarians and the Dutch forgot all about the war (that is, until they fought the Brits again and again in the Anglo-Dutch War, the first of which started just 1 year later!)

In 1985, a local historian and Chairman of the Isles of Scilly Council Roy Duncan decided to look into the rumor that the Isles were still at war with the Netherlands. When the Dutch Embassy in London confirmed that there was an actual declaration of war and everything, Duncan invited the Dutch ambassador Jonkheer Huydecoper to Scilly to sign a peace treaty.

The peace treaty was signed on April 17, 1986, thus officially ending the war between the Netherlands and Isles of Scilly 335 years after it was declared!

Sources:
The Anglo-Zanzibar War of 1896 at BBC h2g2, Wikipedia
Zanzibar Courage at Global Frontiers
335 Year War at Scilly News, Wikipedia
Dutch Proclaim End of War Against Britain's Scilly Isles at NY Times

Quibbles?
Supposedly, an even longer war between Rome and Carthage, which started in 264 BC with the outbreak of the Punic Wars, "ended" with a peace treaty signed in 1985. That would've made it a 2,248-year war, except Carthage was completely destroyed by the Romans and its territory completely annexed. In 146 BC, Carthage's existence as a independent state was over.