I haven't finished the Bleak Falls Barrow quest. But I have attempted repeatedly to purchase land in Falkreath. I am a Thane and there are no more quests available from the Steward or the Jarl. How do I buy land?
To answer the question in the title - the Bleak Falls Barrow quest has no bearing on your ability to get a house in Falkreath. If you are already Thane, the steward should offer you the land, however after reading both Elder Scrolls Wiki and UESP it's not completely clear whether this requires that you are level 9 or not. They both mention that you may receive a letter after you become thane and are level 9 informing you that the land is available.
As mentioned, the house in Falkreath is added by Hearthfire, so you need to have that installed to get the quest to buy land.