When I turn the key, there is a rapid clicking sound*, and the lights on the dashboard and the overhead light all flash in synchrony with the clicks.
Is this a dying battery, or something else?
*Not just clicks, also the barest beginning of “car starting” noise, rapidly over and over again.
Sounds like it. Could also be a corroded battery cable or a discharged battery from a bad alternator, lights left on etc.
Okay thanks.
Clicks is a sign of low battery power. Most likely it’s your battery, but the question remains: Why is your battery dead? If the replacement goes tits up in short order, then the root of the problem is most likely the alternator. Sadly, having drained the new battery with this diagnostic, it can get screwed up and you may have to replace it, again, when you do the alternator. A bad battery can finish off a dodgy alternator, and vice versa.
Clarify.
If it sounds like the engine is turning over very rapidly(like, faster than usual) when the starter engages, that’s your timing belt/chain being broken. Completely unrelated to the battery issue, and a much bigger problem. Last time the car ran, it didn’t “just die on you”, did it?
Sounds like the battery to me.
At the first sign of any starting problem, take your car to an autoparts store that does free diagnostics. They can test the battery, starter, and alternator in just a few minutes. Most places will even install the new battery for you.
No, one night it worked fine, the next morning this.
In this case, there was no sign of any starting problem until the morning it wouldn’t start. 
Sorry for my inexperience here–in 15 years or so of adult life I’ve only had to deal with things like this probably less than five times.
So what I’m hearing, I think, is not just to get a new battery, but to get the car taken somewhere where they can figure out why it went bad. Sounds about right?
-KR
the rapid clicking is the starter solenoid engaging and disengaging over and over. The battery has enough capacity to energize the solenoid and engage the flywheel, but once the starter motor tries drawing power, the battery voltage drops to the floor and the solenoid disengages.
how old is the battery? if it’s approaching or over five years old, it may have just given up the ghost.
Yeah, unfortunately I have to admit I don’t remember how old the battery is. I don’t remember if I ever got a new battery for this car or not. (The car is six years old)
So anyway, I’ll see if I can jump it once I get access to cables, then whether I can or not, I’ll get it transported to the shop, and they’ll take a look while I get shuttled around by my dear wife.
Can someone give you a jump? That will probably get the car started and you can drive to the parts store.
If you don’t have jumper cables but someone can give you a ride, take the battery out of the car and take it to the parts store. They can test the battery and you can pick up a new one. Have you ever removed a battery? It’s not too hard. Checkout youtube for some videos on how to do it. The only really important thing is to disconnect the negative terminal first.
Ummm no.
You can connect the negative first and maybe not have a bad result ( as long as the wrench you have on the positive cable doesn’t touch ground you are golden)
The REALLY important thing is to make you connect positive to positive and negative to negative. Put the cables on the battery backwards and it gets very expensive very quickly. Electronic control units don’t like being hooked up backwards.
And yes I have seen customers do this. I have also seen $2500 repair bills as the result.
I guess that’s true. It is most important that the battery be connected properly and not reversed. But it’s a good idea to disconnect negative first when taking the battery out and connect negative last when installing the battery. This way you don’t risk creating a short if your tool touches the frame when working on the positive connection.
No guessing about it. It’s true.
Yes I’ve removed a battery before. Anyway I’ve decided to delay all this activity until tomorrow morning when my calendar’s clear. Today I’m just stranding my wife at home. She’s being a trooper about it. 
Sounds like the battery. There should be a sticker on the battery with numbers peeled off or scratched out to tell you how old the battery is. As other’s pointed out, it could just be a corroded cable. Also your battery could be fine and it’s not charging for some other reason, though I’m not sure what in a modern car.
Has anyone considered that maybe the battery was going dead?
Depending on the car, it may not be so easy to remove the battery. Some can be very difficult. I’m looking at you 1996 Chrysler Sebring. The battery came out through the wheel well. You had to jack it up/use jack stands or take it somewhere with a hoist as you had to remove the tire to get it out.
The choke might be broke.