9 times out of 10, when this sort of thing happens, it's a problem with the battery cables. Oftentimes it's the ground. Check for corrosion, make sure the terminals are clean - not just on the battery terminal end, but on the other end too.
And lastly, check the solenoid. They more commonly stick 'on', when the starter is engaged and won't shut off, but sometimes they just refuse to engage, too. You can test it simply by using a crummy old pair of pliers or something similar to bridge the main terminals. If there's a big fat spark and the starter kicks, then you can be sure it needs replacement. (just be sure the car's in Neutral or Park if it has an auto!!) IF you have the key in and on 'run', it should start, too. While the low (12 volts) are not going to shock you, don't hold your pliers on the terminals too long, or they will get very hot, and probably weld themselves together, or to the terminals. High amps are weird like that. It's a good idea to wear a glove, just to avoid hot spots.