My G5 Mac with Tiger has been very slow, displaying the "beachball" frequently. I turned it off and when I turned it on again, I get a grey screen, a chime and a blue circular image in the middle of the screen. It doesn't boot up. Do I need a new computer? It is 7 years old.

Well, there are a lot of reasons that can consequently cause the issues on your Mac. Meanwhile, there are certain ways through which you can do to fix one by one. However, if you prefer a more efficient way to resolve, try an all-in-one Mac speed up software as it can help resolve all issues in just a few clicks.

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Sounds like there is a issue with either the operating system or the hard drive.
Try booting from the Mac OS X install disc. insert disc press power while holding down the "C" on the keyboard.
If you can, while running from the disc, go to the Disk Utilities app within the Applications > Utilities folder
select the hard drive and run disk permissions. If it runs without any red errors the issue is with the operating system. If you get any red errors listed or it doesn't finish the job, its a hard drive issue and it will need to be replaced.