Caravan
There is a short sequence of quests that leads to the OXYALE, and it begins with the BOTTLE. You'll find it for sale at a cost of 50,000 G at the Caravan. The trick (aside from having the money) is actually finding the Caravan. You'll find hints to its location in Onrac, but there is no specific indication on the World Map as to where it is found.
The Caravan is located in the middle of a tiny patch of desert north of the much larger desert west of Onrac. You can land south of the desert, past the River and forest, but it's a much shorter trip if you land on the thin strip of land west of the Caravan. You'll fight the same tough encounters either way—watch out for ANKYLOS in particular. (An even quicker way is to land the AIRSHIP directly on top of the Caravan, which is possible due to a bug.)
The Caravan is a simple shop with a single item for sale: the BOTTLE, which costs a whopping 50,000 G. Purchase it and head back to the AIRSHIP. The BOTTLE contains a Fairy whose home is in Gaia, and that's your next destination. When you use the BOTTLE, the Fairy automatically returns to Gaia, regardless of where you are. The Caravan will also become inaccessible at this point.
Return to Gaia
This is the only time you actually have to visit Gaia, though you likely have already due to its convenient location. If you have any money left over after buying the BOTTLE, this is a good place to spend it. Your main goal is to head to the Fairy spring behind town. Follow the road to the west and continue along the small northern path. When you reach the Clinic, head east along the forest path. You'll find the pond at the end; at this point you'll want to use the BOTTLE from your inventory screen if you haven't already. The Fairy will offer OXYALE as a reward if you speak to her. With the OXYALE secured, it's time to return to Onrac.