The Fesarius-class was a type of First Federation starship used in the 23rd and 24th centuries. The class was spherical, and over a kilometer and a half in diameter. Even in the 24th century the class vastly outmatched any Federation vessel; faster and more powerful than even the Sovereign-class.
As was typical of First Federation technology the Fesarius-class used a type of unique advanced warp drive, with specifications unknown to the Federation. They were built by self-constructing robotic devices which were stationed on asteroids to transform the base materials into the vessels, at practically no cost in resources or effort to the First Federation, because of this the First Federation fleet could be much larger than the number of ships they had in use at any one time, with additional ships ready and waiting for future use. (TOS novel: Preserver)
The first Fesarius-class vessel encountered by the Federation was the Fesarius, which made first contact with the USS Enterprise in 2266. Following this contact the First Federation made several more contacts with the Federation before deciding the Federation was insufficiently advanced to continue full contact. Two of these meetings were held at a First Federation asteroid base, which had berths for up to fifty-one Fesarius-class vessels; at the time seventeen were docked. (TOS episode: "The Corbomite Maneuver"; TOS novel: Preserver)
In the mirror universe the First Federation had the same facility, however when mirror-James T. Kirk tortured Balok to try and find its location, the alien refused to reveal anything. In 2375, then Emperor Tiberius travelled to the First Federation base in the primary universe on the ISS Enterprise-E, hoping to capture the Fesarius-class vessels. In the base he found only wrecks of the vessels, which had been destroyed six years earlier by an asteroid impact caused by a Preserver obelisk. (TOS novel: Preserver)