First, 2-key-3DES is significantly different from 2DES (which provides little additional security over DES due tobecause of MeetinMeet-in-the-Middle-Attachs attacks). 2-key-3DES does Encryptencrypt with key1, Decrypt with key2 and encrypt with key1 again (instead of a fresh key).
Now DES indeed uses arbitrary bitstrings as keys (possibly filtered for the know weak keys and a 2-key 3DES uses a pair of these bitstrings which means you'll end up with something close to $\{0,1\}^{112}$.