My time machine backup isn't working. When I open Airport Utility, I see an error message that "Internal disk needs repair." Is this referring to the disk in the Time Machine? Everything else in iMac seems fine.

I believe you are asking about the hard drive in a Time Capsule......(since Time Machine is software on your Mac that is used to back up the data on the Mac).

Unfortunately, the word "repair" here means that the hard drive has failed. Apple will not provide any help in this regard, but a local shop might be able to replace the hard drive for you with another new drive. Cost of the replacement hard drive and labor will be in the $250-300 range in most cases.

Now, if the Time Capsule is 6-7 years old.......the power supply inside the Time Capsule is also about at the point where it can fail at any time, so it would not make much sense to replace the hard drive in this case unless you also have the internal power supply replaced as well. That will move the estimated cost of repairs up to $350 or higher.

Yes, this is referring to the disk in the external backup hard drive. If the external hard drive is made by Apple, you may be able to get some assistance at the Genius Bar. If not, you could still try to see if the Genius Bar can help.

Genius Bar Reservation and Apple Support Options - Apple