The blasts were heard across the city and pillars of smoke rose from the depot, where looting had broken out a day earlier, an AFP correspondent reported.
The depot is at the foot of Jabal Hadid mountain overlooking the port, and a large amount of weapons dating back to the Soviet era were stored in a cave there.
Troops guarding the depot abandoned their posts this week after their commanders fled as chaos tightened its grip on the war-torn country.
Houses shook, windows were shattered and several nearby buildings were destroyed in Saturday’s explosions, residents said.
“We have so far retrieved 14 charred bodies,” health department director Al-Kheder Lassouar said.
“We have information that there are more bodies inside that we were not able to reach” as smoke engulfed the area, he added.
Many people were inside or on Jabal Hadid when the explosions took place, witnesses said, reporting that several had died in the looting even before the blasts.
One resident who was at the mountain earlier on Saturday before the explosions told AFP that “there were many dead bodies in the cave”.
“People entering to take weapons were killed by others who didn’t find what they were looking for,” he said.
Others reported seeing several people die of suffocation after inhaling toxic gases released from gunpowder stored deep inside the cave.
Several others were stabbed, residents say.
AFP could not confirm these reports from official sources as fighting between local militias and Shiite rebels, under fire from a Saudi-led coalition elsewhere, killed 61 people in three days, according to Lassouar.
Later Saturday, local anti-rebel popular committees forces seized full control of Aden international airport, following clashes with Huthis in which both sides used tanks and in which medics said the popular committees lost five men.
Special forces officials close the Huthis said nine men on their side were killed.
President Abedrabbo Mansour Hadi took refuge in Aden after fleeing the rebel-held capital last month.
He left this week for Saudi Arabia before heading to Egypt for an Arab League summit with the Yemen crisis high on the agenda.