Living in Asia, where most people don't have a landline phone, they aren't much fun.
For a while, they were a lot of fun in America. But, like Rowan and Martin's Laugh-In, it was a fad bound to die within a few months.
My favorite line was to record my message:
"Hi. You've got me when I can't come to the phone. Please leave your name, number, and a short message after the tone. Thanxcelot!
And that was supposed to be the end of the message, but I pretended to make a mistake in recording...
"HONEY, WHERE'S THE TV GUIDE?"
Nowadays, if you have Voice Mail, you don't need the machine and that works even during a power failure.