Assuming the largest ships that can be accommodated in the Death Star I's hangars is equivalent to the largest ships in its permanent garrison complement, it is a medium cruiser class.
The following answer is Legends-only.
The Death Star Technical Companion (1991), a book published for use with the Star Wars Roleplaying Game, provided the following excerpts (emphasis mine):
Besides the huge amount of TIE fighters, the Death Star carried assault shuttles for boarding captured ships not deemed safe enough to tractor into a hold, blastboats for use as far-patrol craft, a limited number of small capital ships (specifically strike cruisers), drop-ships to carry troops and equipment to a planet's surface, land combat vehicles, and a wide variety of support craft such as shuttles, cargo haulers, repair platforms, and worker cabs.
Based on the strike cruiser's visual profile, it is identified to be the Strike-Class Medium Cruiser. As described in Wookiepedia, it is 450 x 90 meters in size, and tactically equivalent to half a Victory-Class Star Destroyer. By comparison, a Victory Star Destroyer is 900m long and an Imperial Star Destroyer is a mile long.
Additionally, it must be noted here that no Imperial hangar is ever needed to fit a Star Destroyer of any size because they have no landing gear (also why TIEs can only use purpose-built TIE hangar bays and not any ordinary hangars).
After the Clone Wars, the implementation of the Tarkin Doctrine led to a shift in the military ship design philosophy towards ever larger capital ships for ever greater fearsomeness (the Strike Cruiser capital ship is an exception developed in response to the weakness exposed by this trend during the Galactic Civil War). Star Destroyers of the Imperial era are too large to land on planets (the Victory Star Destroyer is the only class that can operate in the upper atmosphere). Since Star Destroyers are now permanently space-bound, landing gears are removed by default, since they are built in orbital shipyards and operate only in space). The Sentinel-Class Shuttle is designed to take over the Star Destroyer's former role as the Empire's primary planetary troop lander.
With a loss of landing gears, Star Destroyers will require purpose-built hangars like TIEs do, for jobs like ship-wide maintenance and repairs, or movement of supplies requiring faster throughput than shuttles alone can provide. These hangars are probably the "docking" you read of. Not external linkage like here when General Grievous is boarding Obi-Wan's ship:
But docking bays, drydock style, akin to construction bays in orbital shipyards where the Star Destroyers themselves were constructed at places like Kuat and Mon Calamari:
Sure, there's no proof that such bays actually exist on the Death Star. The Death Star I doesn't have a fleet escort (thanks to Tarkin's overconfidence) but they designed the Death Star II to have one. You can't have crippled Star Destroyers limp back to the orbital shipyard and leave the Death Star exposed, and you can't let crippled ships bog the Death Star down while the Rebels are busy evacuating their base a hyperspace jump away. There is a tactical need for Star Destroyers to have the ability to dock with the Death Star, so why reinvent the wheel?