Rob,
The clock is actually not a significant source of battery drain if it is working normally. It fires about one time every 7 minutes and draws about 0.25A for 0.2 seconds; this is equivalent of a steady drain of 0.1mA. Over the course of a day this would be about 0.0024Amp-hours and the battery should be at least 50Amp-hours or more, basically able to run the clock for a decade. Do you hear the clock solenoid firing (sort of a ker-chunk sound) and if so what is the approximate interval? Another note on the clock is that the most likely failure mode is for the contacts to corrode / pit and basically it will stop firing and thus stop drawing current.
If you do have a battery drain I also suspect that it is a light or a short. In addition to the trunk light suggestion there is a glove box light that can do the same thing. A small light can draw well over 0.2 amps continuously; this will drain the battery of 10 amp-hours each day or more.
On the battery how do you know it is drained? If starter is sluggish or not turning over well it could be a bad battery cable or starter connection (or even the starter/solenoid). If these seem fine I would recommend a battery load test to see if it has failed or was defective, many part stores will do this gratis. Also, if you have a volt meter you should look at the voltage when the car is running - it should be between 13.5v and 14.5v. If not, the alternator may not be working properly or the battery has a shorted cell.
If the battery and alternator appear good, my recommendation would be to pull the positive battery terminal. The first thing to do (with the ignition off) would be to touch the positive cable terminal to the positive battery post and look for a spark spark (make sure you are in a well vented area and no fuel fumes or battery vapor is present). If you see a spark the first time it could be the clock firing but if you do it again after a few seconds there should be no spark. Ideally you would then connect the cable to the terminal through an amp meter and look at the current; it should be zero or certainly not more than a milliamp. If you see current, start pulling fuses until it goes to zero and you will find the offending circuit (there are also a few circuits from the battery which are not fused to consider). From there we can go through the wiring diagram and look for possible causes.