I had an uncle who hunted with a takedown 99. He passed away in the early 60s, so it's been a good many years since I've seen it done, but I recall it being quite similar to the interrupted thread system used on the Winchester M97 and M12 shotguns.
If I recall correctly, he pulled down and/or rotated the little release catch, and slid the forearm forward about an inch, then rotated the barrel & forearm about 90 degrees and pulled them out of the receiver.
I never got the opportunity to try it myself, and never really paid very close attention while he did it, as I was generally spit-shining my nearly new Marlin .30-30 at the time.