A friend from high school had this one:
I’m an old man from Nantucket.
I’m old, but I ain’t kicked the bucket.
Don’t be a creep,
Leave your name at the beep.
If you don’t, well, then you can just f. . orget about anyone calling you back.
I think that would be even funnier verbatim. 
My friend’s machine, for a while (in a deadpan voice):
*
"Welcome to the rabid monkey hotline.
If a rabid monkey is currently clawing out your eyes, press 1.
To hear about the latest rabies vaccinations, press 2.
If you wish to speak to the next available rabid monkey representative, please leave your name and number after the beep."
beep
*
Yumanite:
<laugh laugh laugh> ah, God, I’m cracking up
I like this one incorperating Theodin’s quote in LOTR:TT
Fell deeds await… Now for Wrath… Now for Ruin… and the Red Dawn…Forth Eorlingas! Leave your message at the beep!
A friend of mine simply had “Answeringmachine” as text. No name, nothing. Since it was spoken in his usual, slightly lunatic kind of way, I thought it was awesome!
As for TV answering machines, I liked the one Nick Knight one (you know, that Canadian police vampire story):
“Either I’m in bed or don’t want to be disturbed… Or both!”
And from this thread, I absolutely loved the one “I’m currently out, making a difference!”
P.S.: Yet another ancient thread that tricked me. Damn, I really, really need to check the timestamps 
I heard a cute one the other day. Two little kids, sounded like they were about 6 or 7 years old.
“Hi, this is Jack” [and in background] “I’m Mitchell”.
“Mummy and Daddy can’t come to the phone just now” [and in background] “We’ve tied them up under the stairs” [much giggling].
“But if you want to leave a message, wait for the beep…BEEEEEEEEEP”
Pissed myself laughing I did. 
You’ve reached Fuji. I might be here, but I screen my calls. You have exactly thirty seconds to justify why I should get up off the couch and answer your call.
My boyfriend’s message…
Phone rings a couple times.
“Hello?”
pause for about 3-4 seconds
“Haha, you suck. Not here right now so leave a message…”
We got my wifes parents a new answering machine for a gift. when they were setting it up I BEGGED them to leave the first message they recorded. They wouldn’t, but it was classic.
It went something like this.
Mom: …ays to push and hold the record button until the light blinks.
Dad: I did that.
Mom: When the light blinks that means its recording.
Dad: Which light?
Mom: This light is blinking. Is that the one?
Dad: I don’t know YOU have the book.
Mom: Well dear I…
BEEEEEEEP!
Our friends did the “If you’d like to leave a message for Ray, press 1 now. If you’d like to leave a message for Caroline, press 2 now.” many years ago before these call routing systems were ubiquitous. Great idea, except that Ray had never changed the call retrieval code on his answering machine, so the first befuddled person to actually follow the directions got to listen to all their incoming calls.
His message went something like: “Uh, hi Ray. I called to let you know that we can’t make it to the restaurant til 7:30, and btw, your mom wants you to drive her cat to the vet next Tuesday, you stupid idjit.”
(Steel drums) Thank you for calling Carnival Cruise Lines. There has never been a better time for a luxurious vacation to beautiful Haiti. Discounted travel executions are now available. Pay no attention to the truth less rumors of unrest, there has never been a better time for a luxurious vacation in beautiful Haiti…
Old Message:
Dog:scratch scratch WOOF!
Current Message:
Press 1 to be ignored by a human!
Press 2 to be ignored by a machine!
Press 3 to talk to the dog!
Please press a number now!
10% of callers press a number