Does anyone know if it's possible to customize the voicemail greeting on iOS 17?

It has always been possible to customize the voicemail greeting from an iPhone. Note that the greeting is stored/used by your cellular carrier as it needs to be used when your phone isn't able to be contacted (e.g., no cell service, phone off, phone broken, etc)

Open the Phone app, tap Voicemail at the lower right of the screen, tap Greeting at the top right of the screen, tap Custom.

Also see: Set up your voicemail on iPhone - Apple Support

FabioFZero wrote:

This option doesn't show up for me. It's a bit weird this requires carrier support when it's actually handled locally by the device. It doesn't make much sense.

FoxFifth is correct. Voicemail is a carrier feature. Voicemail is not local to your phone. However, the carrier does provide settings to Apple that control how you can access the features.

There have been some reports since the release of iOS 17 that people have been having trouble accessing the greeting features. I don't know which carrier. I suggest you visit your carrier's website and find out how to set up a greeting directly through their system. With my carrier, Verizon, you dial *86 (VM), interrupt the recording with the # sign if necessary and then you'll be taken to the settings menus. That may work for your carrier or it may not.

A call goes to voicemail when the device (iPhone) is not available to answer the call. Even if your phone is turned off and then run over by a bus, people can still call your number, hear the greeting and leave you a voicemail message and you can retrieve that message from another device, including a landline. No iPhone involved.

Not all carriers support Visual Voicemail and other features. If your cellular carrier doesn't support the features, you should be able to set up a custom greeting directly with them.

Have you checked with your cellular carrier?