Many classes can solo, but the ones that stand out with the most potential are Beastmaster and Dragoon.
Beastmasters can Charm their prey and force it to fight for them against other prey alongside them.
On the other hand, Dragoons use their wyvern's ability to change support systems through a variety of subjobs to solo. When subbing Black, White, Blue or Red Mage, the wyvern becomes a support wyvern and uses the ability "Healing Breath" depending on the health of the master or nearby party members whenever the Dragoon casts a spell. They can further increase the usefulness of this by using their Artifact helmet.