With patch 2.1, Free Company Housing has become available. Any FC that has reached rank 6 is eligible to purchase a house, although it can be quite costly. Currently owning one has no benefits, but in the future it is expected to be added.

Unlock housing tip
Before you can even think about buying a house, your Free Company must reach level 6. This can be done by simply doing more of what your FC has already done in the past. Leveling, completing quests, leves, these all add to your progress towards ranking up.

Upon reaching 6, you can then go to one of the three available locations outside of each starting town. With money in hand, you can then walk around, browsing the different lots and then purchase a plot of land. With the plot, you can then set out and begin building your house at your leisure.

The housing zone location
Housing has been given it's own zone, and the new zones can be found attached to the low level areas outside of the starting cities.

Mist - Lower La Noscea - X34,Y19
Lavender Beds - Central Shroud - X27,Y26
The Goblet - Western Thanalan - X25,Y27

If walking isn't your thing however, you can complete a quest in each starting town that will allow you to use that towns Aethernet to teleport directly to the housing zone.
Where the Heart Is (Mist)- Lower La Noscea:Ahctkoen - X32,Y20
Where the Heart Is (Lavender Beds) - Central Shroud:Margeria - X21,Y22
Where the Heart Is (The Goblet) - Western Thanalan:Imme - X25,Y24