The offices of the European Union and the High Peace Council are also nearby, along with a number of foreign embassies and the police headquarters.
In a message on Twitter, the group said more than 50 wounded had been brought in to that hospital alone.
A spokesperson from the public health ministry said several people had been killed and many wounded in the blast.
Zabihullah Mujahid, a Taliban spokesman, claimed responsibility for the attack, which sent thick, dark smoke into the sky from from the site of the explosion near the government's former Interior Ministry building. The Council was set up in 2010 to negotiate with the insurgent group.
Smoke rises after a vehicle bomb explosion in Kabul on January 27, 2018.
In chaotic scenes at the Jamuriate hospital, which is the nearest medical facility to the blast, overwhelmed doctors and nurses rushed to treat dozens of wounded lying in the corridors.
Outside the hospital, people were carrying the injured on their backs and paramedics loaded several people into ambulances at once.
"It's a massacre", said Dejan Panic, a coordinator with the NGO Emergency, which operates at hospital in city that treated victims.
The blast comes a week after gunmen attacked the Intercontinental Hotel in Kabul, killing at least 22 people during a 12-hour standoff with security forces. He said he didn't know yet if any of their members are wounded or killed.
The explosion ripped through Kabul shortly before 1pm on Saturday afternoon and shook building kilometers away. In another attack on the Afghan soil, an office of the Save the Children aid agency in Jalalabad had been targetted by militants wounding 11 people, reported Reuters quoting local sources.
Smoke rises after a auto bomb explosion in Kabul.