Help, how high am I?
Solution 1:
You can find (from killing Cave Bats, with 0.5% drop rate) or build (if you're playing on console) a Depth Meter that will add that value to your UI. (And also a Copper Silver or Gold watch if you want the time shown in that screen shot)
UPDATE: In 1.1 a new item called the GPS was added that combines the functionality of these two items plus a third that tells horizontal positioning.
Solution 2:
You can craft a Depth Meter with: 10 Copper Bars 8 Silver Bars 6 Gold bars
You need to craft it at a Table and Chair, which is just a chair facing a table.