Allod construction is a series of quests for those that have unlocked greatness and goes through several stages.
Part 1 is relatively easy and requires a minimum of 5 weeks. You need to collect 100 Radiant Astrilium from the last boss on an upgrading sector(S6 and S8+) allod. Alternatively you can hand over a key and 200 gold to hire an NPC to get one for you.
Parts 2-5 are about earning earning meteorite fragments which come from dailies and a weekly. Each part takes 4 weeks minimum and the amount of fragments needed per week in each increases with each part(2 is 10, 3 is 14, 4 is 18, 5 is 22). The quests appear to rotate in a cycle like the quest for speeding ship construction/upgrade/repair. Once you have enough fragments to do the weekly turn in the daily quest NPC will stop offering new quests. You can still complete and turn in any quests in you quest log though.
- Solo quests(1 fragment)
- Kill 50 elementals in Kingdom of Elements with your flag of war on.
Earn 15 Symbol of Achievement from Cradle
Tka-Rik 10 trial set.
Get 1st-3rd place in Wandering Island or Kirah race.
- Open the race or battle chest from June Catacombs.
Open a chest in Melting Island.
Win Witch's Hollow.
recover over 50% of chests from AoD
- Kill an Ancient Octulhu in a Dark astral sector.
Kill the last boss on an allod in a Brilliant astral sector or deeper. Anything that does not have dull in the name will work.
- Kill one of the 3 Dead City wing boss.
I will edit this as I get more information.
Edited to add information from v_buster
Edited for corrections.