http://forum.square-enix.com/ffxiv/t...-Eureka-Anemos this seems to have full information that you need. I've had a glance through it and it is extremely detailed, if you need the full details, I will however give you an overview.

Eureka is basically an open world instanced dungeon (finally!), in which the mobs are all out to eat your face. You will need to first of all obtain a Protean Crystal (you obtain this by killing mobs, it's RNG based though, getting a chain kill of enemies does help. Do note that killing enemies below your level will result in no chain being granted, and no chance at a protean crystal).

Once you have this, you then talk to Krile again and hand it in. At this point you will then be able to convert Anemos Crystals (earned from FATEs/NMs/whatever we call them) to Protean Crystals by speaking with Gerolt. You will need to use Protean Crystals for everything except the final Anemos stage (i355 weapon/shield, i350 armour). For the final stage, for your armour you'll need to use Anemos Crystals, for the weapon/shield you'll need Pazuzu Feathers, obtained at level 19 and 20, from the Pazuzu FATE. Be advised you can lose levels from dying (this is negated if you are resurrected).

There's a low-down, use the link for a full guide. Hope this helps you get started.