No light and AirPort Time Capsule not recognized by Mac

No light = no power on the Time Capsule, likely because the internal power supply has failed.

You might not be aware that when the Time Capsule is disconnected from your Mac, that Time Machine will make "snapshot" backups that are stored on your Mac, so that is likely what you are seeing.

Unfortunately, if you need to completely restore your Mac, you will need to be able to access the complete backups on the Time Capsule. If the Time Capsule is 5+ years old, repairs are probably not feasible. You can have a shop open up the Time Capsule and pull out the hard drive. Then, place the hard drive in a caddy or enclosure and you should be able to access the backups stored on the Time Capsule hard drive.

We'll guess that you have a 4th Gen Time Capsule, since they were notorious for disk spinning, but no power light and no accessibility from your Mac. Part of the power supply is working and part is not. This is a usually board solder issue that can often be fixed by "baking" the board to reflow the solder. A really good do-it-yourself type of person experienced with small electronic repairs could probably handle the process.

Check to see what model of Time Capsule that you have. If you are not sure, look on the bottom of the Time Capsule. The model number is usually embedded in the foam pad base.......starts with an "A" followed by 4 numbers.

A 4th Gen Time Capsule is model number A 1409. This model was discontinued 8 years ago and is labeled as "obsolete" as far as service status from Apple, meaning that they will not help you with your Time Capsule, even if you are willing to pay them.

I still think the best course of action is to pull the hard drive from the Time Capsule and place it in a caddy or enclosure so you can access the data on the drive that way. Make a complete copy of the hard drive as soon as possible, as you would not want to keep data that is important to you on an 8+ year old hard drive.

Instructions for oven bake are here.. assuming as Bob has suggested.. your TC is Gen4. A1409

https://sites.google.com/site/lapastenague/a-deconstruction-of-routers-and-modems/apple-time-capsule-repair/gen4-a1409-issues

IMHO the day of reckoning has come. A dead TC could be got operational for another year?? More maybe.. but without a couple of spare Gen4 TC I would say the time has come for a new router and new Time Machine backup target.

Products of the TC age.. now 8 years.. are well and truly past EOL. Plus Big Sur and Time Machine don't make a great combo. So if you are or will be running Big Sur it is time to move.. NOW!!

I think it's just Murphy's Law, as I need the Time Machine as all of my security cameras are connected through it and I have to go out of state on Saturday.

We have no idea of your current network layout.. but if there is an ISP modem router in the network just buy a switch.. they are cheap as chips.. and available from any computer store. Buy an 8 port gigabit any brand.. they are pretty much identical.

If your cameras happen to be POE then buy the POE version switch and you can get rid of a mass of POE adapters.

Otherwise .. if you need a new router (since TC was in router mode) you can buy a TP-Link cheap one just to fill in.

Most people are moving to mesh if you have any decent size house and a bunch of wifi stuff.