Could be anything from a bad compressor, coked condenser, bad start relay or bad capacitor. Not all ref compressors use a start capacitor.
Unplug and allow compressor to cool off. Make sure condenser isn't full of lint before you try to restart it. If it fails to restart try the following:
First thing I would do is remove start relay/or start cap. There are 3 terminals on compressor. Common, start and run. Measure the resistance from common to start and common to run. The exact resistance isn't important. Need to make sure wires aren't open. Will get a resistance of 10 ohms or less. There should be a diagram or marking someplace to determine C, R and S. Also check wires for shorts to ground, the metal of compressor.
Second, I would look for a capacitor. Many old ones didn't use start caps.
Third, If compressor didn't have a start cap, I would rig up a cord to connect power to common,C, and neutral to run ,R, winding.
Fourth, use an amprobe to measure current on common leg.
Fifth, Connect a jumper wire from run to start. I would use a short jumper with insulated alligator clips so they could be removed easily.
Plug in the power cord you made up and a second later remove jumper from start to run.
Compressor will then run on the power connected to common and run.
If compressor uses a capacitor you will need to connect it with a jumper between run and start and remove cap a second later too.
If you don't have a clue what I'm telling you, CALL A PROFESSIONAL OR GET A NEW REF.
The clicking sound you are hearing is the Klixon opening. You can google Klixon motor protector.
My dad was in the heating, air and ref business. So you could say, I grew up working on this stuff. If you don't feel comfortable with electricity, call someone.