Let me just get this said: if #18 is the most "gleeful anachronistic picnic", the most light-hearted use of a completely anachronistic theme, of any episode: then #23, in my humble opinion, is the most serious. More than in any other episode, anachronism --and extreme anachronism at that--is used here for political and social satire.

Please Note: it is a major mistake to date Champloo in the 1800s based on this episode, as we will now show.

Let's start with the biggest one:
The United States of America. Put very simply, it didn't exist yet. Champloo's core date of 1675 (see many examples of this at Dating Champloo) is almost exactly a century before the 13 Original Colonies would fight their war of independence from Great Britain. The major concerns of the Colonies --not yet states--at the time were small, internal disputes: a war between the New England colonies and the Wampanoag Indians (King Philip's War, 1675-78), and the short-lived insurrection of the Virginia planters led by Nathaniel Bacon (Bacon's Rebellion, 1676-7). There was not yet any such thing as a "United States of America", let alone a military force representing it, and there was most definitely neither any thought of interfering in other countries' affairs (beyond Britain, with whom there were already some quarrels) nor the power to do so.

The Amekou (Americans) in the episode look as if they come from the 1800's, rather than the 1600's. (The sailors' uniforms are a close match to American Naval uniforms of the period; ordinarily they would be dark blue rather than white, but they may be wearing summer uniforms because of the warmer climate. The baseball uniforms are also quite late-1800s in style. See photo below.)
-- This makes relative sense, as the 1800's are the period in which:
(a) The United States first actually did make contact with Japan and
(b) Baseball was invented.

Taking these in order:

Commodore Perry and the "Black Ships". I am openly and gratefully swiping AnimeForever's note here, since I don't think it can be improved on, and I offer my thanks to bebop-aria and cowgirlnoir, who probably wrote it:
"In 1853, American Naval Commodore Matthew C. Perry first visited Japan, requesting that the Tokugawa shogunate allow trade between their nations. He was rebuffed and returned again in 1854 with a fleet of nine war ships, now commonly referred to as the Black Ships, or 'kurofune'. As a result of this move, a treaty was signed at the port of Shimoda allowing trade at that port and at Hakodate. Perry's brute force overture may have helped the end of the Tokugawa shogunate's two-century policy of isolationism, which was formally eliminated during the Meiji Restoration in 1868."
--Cartwright and Doubleday make a very similar "brute force overture" here: armed to the teeth, they directly state that a refusal to open trade relations will be considered grounds for immediate attack ("Make no mistake, this is not a request: this is an order"). Note that the response they are given is the one appropriate to the period of the Tokugawa isolation, namely that all foreign ships are to be driven away. --It's almost certainly intended as a reference to Perry's tactics, as well as to American attitudes toward world affairs in general since WWII.

===And to make it even more pointed, Adam Randis brings to my attention that while the episode's Joy Cartwright is obviously named for the real one, his physical appearance is almost certainly based not on Cartwright but on Perry--compare the two images above.

The game of baseball. Which also didn't exist yet. According to History of Baseball: "Alexander Joy Cartwright (1820-1892) of New York invented the modern baseball field in 1845. Alexander Cartwright and the members of his New York Knickerbocker Base Ball Club, devised the first rules and regulations for the modern game of baseball. Baseball was based on the English game of rounders. Rounders become popular in the United States in the early 19th century, where the game was called "townball", "base", or "baseball". Cartwright formalized the modern rules of baseball. The first recorded baseball game was in 1846, when Alexander Cartwright's Knickerbockers lost to the New York Baseball Club." (The photo shows the Cincinnati Red Legs, the first team to wear knickers rather than long trousers in their team uniform; photo from 1868.)

Admiral Joy Cartwright in this episode obviously represents his baseball-related counterpart. It's also been speculated that W.D., his translator, represents Abner Doubleday, the man often erroneously referred to as the founder of baseball; AF-F points out in its excellent notes on this ep (yes, they're all great =) that Doubleday got this honor based on one piece of unreliable information, and that the act may well have been an attempt to dim the sport's British origins (see update below) --after all, it was less than a century after the War of Independence--by crediting it to an American war hero, which Doubleday actually was.

Update, July 2007:-- It's been brought to our attention that David Block's 2005 book "Baseball Before We Knew It: A search for the roots of the game" unearthed several games very similar to baseball played elsewhere in the world earlier than the supposed American invention, including the following: "The first known record of the term 'base-ball' in America came in a 1791 ordinance in Pittsfield, Mass., that prohibited ballplaying near the town's new meetinghouse. But this was not the first appearance of that name in print. That distinction belongs to an English children's book of 1744, A Little Pretty Pocket-Book. Beneath an illustration of three players in a field, a verse described an embryonic version of the game we consider our own: 'The Ball once struck off, Away flies the Boy To the next destin'd Post, And then Home with Joy.' Substitute 'base' for 'post,' and you have something that sounds awfully familiar. By 1796 the rules of this game were well enough established for Johann Gutsmuths, a scholar in Schnepfenthal, Germany, to describe them in a book about popular games and sports. Gutsmuths's 'englische Base-ball' is clearly recognizable: two teams face off, with a pitcher serving to a batsman who has three shots to put a ball in play before attempting a circuit of bases on his way home. It also predates the rules laid out by the Knickerbockers by nearly 50 years.'

--so the sport may indeed be older==and more definitely British==than assumed, though this is a source unlikely to have been known to the Japanese.

"The sport of gentlemen". You can find everything from cricket to horse racing to golf, fly fishing and even boxing (!) dignified by this gracious label, but I cannot find any reference to baseball being so described. The Amekou are just being elitist again.

(Note: I have looked carefully, but have seen no use of an American flag anywhere in this episode. If anyone spots one, do let me know where it is; but I suspect there isn't one. It would be interesting to see if they'd have used a design besides the modern--i.e., since 1960--13-stripe & 50-star Old Glory.)

Patriotic music. As the USA didn't yet exist, neither did the trappings of American patriotism, making all the military marches played in this episode likewise anachronistic. The first and most notable one used, "The Stars and Stripes Forever", wasn't composed until 1896.

Kagemaru's fancy shades. Another item in the Champloo Collection of distinctive non-Edo-period eyewear. Dig the yellow lenses. =) You can check the Anachronisms Guide listings for Episodes 1 and 4 to read more about unlikely glasses elsewhere in the series.