What would make my Olds Intrigue fail to start?
 

I have a 2001 Oldsmobile Intrigue. Yesterday hubby drove it to work, then when he got home I ran to the corner store, and it was fine both times. About 45 minutes later he left to leave on an errand, and it wouldn't start. It doesn't even seem to be trying. There isn't any engine noise to speak of when we turn the key, just a faint clicking that briefly comes from (or so he thinks) the fuse box. He checked several fuses, which seemed fine. He has a battery charger, which also doubles as a jumper, and the battery seemed fine.

There are some pretty bizarre things happening which I hope will help to pinpoint the issue. We bought the car used about a year and a half ago, and almost immediately the cassette player ate a tape which it now will neither play nor eject. If you hit the eject it will attempt to do so, but otherwise nothing happens with the player. Now (since yesterday) when we REMOVE the ignition key the cassette player will start making sounds like it is trying to eject the cassette. Plus, I have a keyless entry keychain for the car. Since yesterday it will no longer lock or unlock the doors with the keychain (although I hear a click as if it did,) and when I try to do so the H/C needle jumps briefly on the dash and the cassette player starts making that trying-to-eject sound. Normally the cassette player doesn't do anything unless we hit the eject button, so this is quite strange.

What could happen suddenly, without proceeding incident, in a short time, disabling the car from even trying to turn over, and playing exorcist with my cassette player and keyless entry keychain like that?

Based on your symptoms, I think there are 3 likely causes.
Most likely, (and cheapest and easiest to check) is your battery is worn out. A battery can show 12 volts on a charger/checker, but still be unable to put out enough power to turn the starter. The low voltage will make the car's electrical system do lots of wierd things, also. I would buy a new battery before checking the next two items.

Second, you may have corroded battery cables. This would cause all the same symptoms as a dead battery. Check under the boots on the end for flaky, green or white powder.

Third, the ignition switch itself may be worn out/bad. GM has a tradition of making ignition switches that wear out in about 10 years. When that happens, you get lots of strange electrical problems, that are not consistent and come and go randomly.
Let us know what you find out.