While Link can unlock this quest fairly early on, the Akkala region is far off at the very far northeast corner of the map and can be quite dangerous during the early game. This quest is not recommended until Link is more experienced and can safely navigate to the Akkala region. Link will need to head to the Akkala Ancient Tech Lab, just just up the road from the East Akkala Stable.
When you arrive at the tech lab, Robbie will want to see your scars. Remove all your armor so that he can see them.
He will then set you to get the blue flame to power the tech lab. This is very similar to the quest that was done earlier to power the Hateno Ancient Tech Lab. However, unlike the previous objective, the path back to the Akkala Lab is full of monsters. While you can quickly paraglide over to where the flame is located, it is probably a good idea to walk around and clear the area of all the monsters before dealing with the flame. Once again use a Torch to carry the blue flame, following the path of unlit lanterns. Keep following until you reach the furnace back at the Akkala Ancient Tech Lab.
Keep in mind that Akkala is notoriously prone to rain and thunderstorms, so it's recommended to light as many torches with the blue flame as possible.
Once the furnace is lit, this will unlock the travel gate, so Link can quickly warp over to the tech lab. Inside, speak with Robbie to complete the quest and he will give Link three Ancient Arrows. Additionally, Link can now purchase some Ancient gear from Cherry, the statue in the middle of the lab.
Arrive at the Akkala Ancient Tech Lab. Speak with Robbie. Arrive at the Ancient Furnace. Light the Furnace outside of the tech lab. This creates a travel gate to the tech lab. Robbie will reward Link with three Ancient Arrows.
|Purah told you that any ancient materials you find can be turned over to Robbie. You can find him presiding over the Akkala Ancient Tech Lab.|
|You met Robbie, the director of the Akkala Ancient Tech Lab.|
However, Cherry-or rather, the ancient over-isn't working at the moment, meaning it can't create ancient soldier gear for you.
All it (she?) needs to be up and running again is for you to light the furnace outside with the blue flame.
|You lit the furnace at the Akkala Ancient Tech Lab with blue flame. The ancient over is now up and running again.|
It's time to go tell Robbie.
|Thought Robbie initially presumed that it was the power of love that set the ancient over right again, he soon realized the cause was actually you lighting the furnace with blue flame.|
If you give it some ancient materials and a generous supply of rupees, the ancient oven will provide some ancient soldier gear in exchange.