My family emigrated to the United states in 1894 from Hesse. Verbal (loose) history says it was because of something political. I haven't been able to find any ships records for my family which backs up the idea that they came here secretly.
I had a quick look in David Blackbourn's The History of Germany, 1780-1914: The Long Nineteenth Century. tl;dr I don't know of any national or Hessen-specific larger turmoil, but if they were socialists or Jewish, they might have had good reason to emigrate.
So, 1894 was actually a year of significantly less migration than 1893 (which was the tail end of a 1880-93 spike in emigration - reaching 221,000 in 1881 alone), according to K.J. Bade's Vom Auswanderungsland zum Einwanderungsland. Most German emigration was for economic reasons though (often by the rural poor moving out of overpopulated areas after economic crises), so that's probably unrelated to your ancestors. It could have been a factor in their decision to come to the USA though, as the significant German subculture in America had strong ties as a community, chain migration was common, and the "uncle in the USA" was a common cultural motif in Germany.
It wasn't a particularly tumultuous period, and I'm not familiar with any Hessen-specific crises. During the Kaiserreich, Germany left behind much of the turmoil of the early and mid 19th century, which saw revolutions, counter-revolutions, assassinations, beginning of industrialisation, and significant reshaping of the political landscape (notably unification under Prussian primacy). The 1890s saw Bismarck leave power, strong economic growth, and the Reichstag building was opened in 1894.
As it was a time of economic growth and increasing union power, the Hessen history portal mentions increasing numbers of strikes. The Sozialistengesetze (anti-socialist laws) were revoked in 1890 after the socialist SAP (later SPD) continued to gain more and more seats despite being severely restricted, but only in 1888 Bismarck had attempted to pass a law to strip socialists and social democrats of citizenship. Hostility to socialism was rife from all quarters, ranging from conservative rural voters up to Wilhelm II. personally. If your ancestors were socialists or social democrats, they might have sought refuge in America. However, it was probably more typical to emigrate for political reasons in the post-1848 era.
What I did find was that in the 1880s-90s, anti-Semitism was resurgent, and in Hessen specifically the Anti-Semitic People's Party (later German Reform Party) was formed and elected to the Reichstag, founded in 1890 by the Hessen antisemite Otto Böckel. Their electoral base was rural, farmers or craftsmen. While it and a lot of anti-semitic parties never gained any large amounts of seats, Blackbourn states that they infected the general political climate and spread anti-semitism. Further, the Dreyfus affair began in 1894, which might have had an influence on the social climate. If your ancestors were Jewish, this might have been significant.
Do you have any other information on them, like religion, class, education, professions, political affiliations, whether they came from a city or the countryside?