The explosions occurred yesterday at the facility in Primorskiy, near the Black Sea, the region's health minister Tamaz Tsakhnakia told the agencies.
Nineteen holidaying Russians were among those who needed hospital treatment, most of whom had been hit by shrapnel.
All the injured were in stable condition, Tsakhnakia added.
The separatist province's interior minister Aslan Kobakhia confirmed no one had been killed at its munitions store on the shore of the Black Sea.
Abkhazia is internationally recognised as part of the tiny ex-Soviet nation of Georgia, but has claimed independence since a conflict after the collapse of the USSR.
Moscow has thousands of troops stationed in the regions in what Georgia calls a military "occupation", and supports the two provinces financially.