Time Machine is a software program on your Mac that is used to back up the data on a Mac to another hard drive.
A Time Capsule is hardware.........a Wi-Fi router with a built in hard drive........designed to store backups of your Mac(s).
It sounds like you are describing an issue with a Time Capsule that is failing due to its age. An educated guess would be that the Time Capsule is overheating and shutting down. When the product cools down, it can be started up again, but as soon as the product begins to overheat it will shut down again, so the process repeats itself.
Unfortunately, Apple does not offer service or repair parts for the Time Capsule, so there is not really much that you can do at this point other than replace the product.
If you have an Apple Store in your area, my suggestion would be to take the product in to let the Apple folks have a look at the Time Capsule and verify the issue for you.
Replacing the Time Capsule with a similar product is going to be difficult, since Apple was the only manufacturer to offer a product that is a Wi-Fi router with a built in hard drive.
There are a few manufacturers.......Synology and Asus for example.......that offer a Wi-Fi router that will allow you to connect a USB hard drive to the product. Spectrum might offer a similar type product as well, so my advice would be to check with Spectrum first to see what kind of replacement product they might offer.