does a telephone answering machine work in a power outage?

depends , does it get power from a wall plug or the phone line !!

I used to have one with a backup battery, but I'm not sure it recorded new messages during blackouts.

as others have said, some but very few are battery powered.  Check the base station to see if there's batteries.   AND if you've got voice mail through your phone company--(no actual answering machine in your house) that will keep working.  If your landline is important, you need a non-cordless phone OR a phone with a battery in the base station for it to work if the power is off.  If you don't have a corded phone for emergencies, you can usually find them at goodwill for less than $5 or you can buy a new one for under $20 at wally's or best buy.  Check it out--I think (not sure) that you might have to pay for shipping at one or the other of them. 

I have never seen a answering machine that would work without AC power.

But if your AC is out that would be the LAST of your concerns

Mine does not.

Turned power off to move a plug socket and it went off.

Why?

Because it is AC powered of course!

Not if it is powered by squirrels. 

Some digital ones that had backup battery power.  Often these were ones that had a cordless phone and used it's battery while in the base.  Use a UPS and all will work. 

i dont think so

Only if it has battery backup. Also, there's a problem when your phone is on your fiber-optic router which needs ac power to work. Then there can be no telephone access. If your phone service is carried over twisted wire, it will work during power shortage.

If it requires AC power, of course not.