Can I take the hard drive out of my Time Capsule and install it into a different enclosure and, have Time Machine take off backing up right where it left off?

No, Time Machine will start all over again with a complete back up of your Mac and then move forward.

At least, that is what has always happened when I've tried to do this in the past. The reason......Apple Support advised that Time Machine will "see" a different backup destination than it did previously because of the enclosure or caddy change.

IF.....the hard drive in the Time Capsule is 5+ years old, i would not recommend trying to use the Time Capsule hard drive for future backups. Simply add a new USB drive directly to your Mac and start new backups that way. Keep the old Time Capsule hard drive / enclosure around in case you need to go back to an older backup. Very few users ever need to do this though.