Time Machine "Backup Failed; There isn't enough space on "(external HD)".

Can I set up both Time Capsules to be running wirelessly, with one as my wireless internet router, and the other simply being set up through Time Machine as my alternate backup, so then Time Machine would alternate backing up my computer to the two Time Capsules?

Sure, you would simply set up your Mac to back up to both Time Capsules. Time Machine will automatically alternate backups between the two Time Capsules......assuming of course that there is adequate free space on the Time Capsule to handle the backup.

Or would one problem with this "solution" be that my wireless internet connection would be degraded by interference from the second Time Capsule?

No, assuming that the second Time Capsule is not interfering with your network at this time. Remember, your Mac will be backing up to each Time Capsule separately, not at the same time.

I thought that Time Machine is supposed to automatically delete old files to make room for new ones. Is there a work-around? (I do share the Time Capsule with another user's Mac in my home.)

That is the theory, but unfortunately the theory does not work all that well in practice.

Time Machine will usually not have much trouble automatically deleting older small "incremental" backups, to make room for new small incremental backups.......but.....when the Time Capsule hard drive is close to being at capacity.......and......the backup is larger than a normal small incremental type backup.......as can occur after an operating system update or a user has added a lot of new images or media to the Mac.......Time Machine cannot delete enough old data to make room for the new larger than normal backup.

When this occurs, you have two options:

One.....if you need all the old backups from weeks, months, and even years going back in time.......(few of us ever do).......you can copy or "archive" all of the data on the Time Capsule hard drive over to a USB drive connected at the USB port of the Time Capsule. Once the archive copy is complete, then you can erase the Time Capsule hard drive and start all over again with new backups.

Two.....If you don't really need all the old backups, you can simply erase the Time Capsule hard drive and start over again with a new backup of each Mac and work forward from the present time.

Unfortunately, it is not possible to combine the two hard drives on two Capsules to create more storage space. Each Time Capsule stands alone as a separate hard drive.

Now, if you are saying that your first Time Capsule is out of space and the second has plenty of space, you can set up backups from your Mac to go to the second Time Capsule. If you do this, Time Machine will start all over again with a complete new backup of the Mac, then move forward with incremental backups after the first backup is done.