Sue H attened the Midwest US gathering at Anne's and dove into the Bin Temari process - she summarizes the steps for us here:

Anne W. kindly guided me through making a bin-temari at our first mid-west stitch-in.  She has done loads of them, in varying sizes, so she may have some other tips - but here is my interpretation of her directions...

Starting with some sort of thick "slippery" nylon/polyester/rug-like yarn, string, rope... whatever, ...while holding the inner "tail" (so later you can unwind from the center out - like a skein of yarn)  loosely wrap this cord/yarn to create a center core.  This core should be about 1cm circumference less than the external measurements of the container you will be putting it into.

Still keeping the inner tail hanging out, "bag" wrap the inner core in a piece of plastic (half a garbage bag works well).... "bag" wrap is my personal explanation for putting the cord core in the center of the plastic, then gather the edges of the plastic around the tail.  Use a small piece of string, thread or twist-tie to hold the tail/plastic ends in place.  Poke a few air holes in the plastic so you don't trap a lot of air.

Continue with thin yarn and thread to create your mari taking special care to wrap in close and tight around the tail area. (be sure to leave your tail hanging free)  A good build-up of thread and yarn around where the tail enters the mari will give you lots to work with at the end when you will nudge these threads to cover the opening.

Mark and stitch your pattern.  I found one of the trickiest parts is to have the tail/hole in an open area of the pattern design... it's difficult to visualize NOT having the tail as part of a pole.  Also, it is good to try and choose a pattern with small stitches - this will help hold your mari together when it is collapsed.  Another hint is to try and keep your stitches shallow - if you go too deeply with your needle, you may catch some of the inner core, and that will cause havoc when you go to remove the core.

Remove all stickers and clean off all smudges and finger prints from your container - especially the interior.

Remove the thread/string/or twist tie from the tail/plastic.  Carefully spread the plastic out like a collar around the core hole and begin pulling the tail.  If you encounter any knotted areas or tangles, use a thin metal crochet hook to reach inside and un-snag the tangles.  (easier said than done!)  Do NOT remove the plastic liner yet!

Once the inner core is removed, gently slip the mari through the opening of your container.

Using blunt-ended chopsticks, bamboo skewers, unsharpened pencils, etc. and batting, carefully re-stuff the inner core cavity with batting.  The plastic
will help you here working like a funnel for the batting.  Take it a little at a time, making sure that you work the batting into the sides of the mari. If done correctly, your mari will eventually have the firmness of a traditional mari.

When your mari is as firm as you can get it, VERY CAREFULLY snip away the plastic liner close to the surface of the mari - be careful not to cut any of the threads.  Tuck the raw edges of plastic into the opening.

Now, using either pointed bamboo skewers, knitting needle, or other such object, gently ease the outer threads over to cover the opening.  Take your
time with this step - ease and nudge the threads, don't pull or force.

Some people like to finish off a bin-temari by making a crocheted doily, or creating some other cover for the top the container...

Other hints and tips:

From Sue H: I am feeling pretty good with my second attempt here... Anne had suggested to use rug yarn or a thick yarn as the inner core.  Having personally experienced having even that heavy yarn tangle up inside making it difficult to remove, I thought why not try nylon rope like we use on our Scout tents and such... Wal-mart had a nice hank of it to use for closeline in their dollar bin, so I grabbed it. (thought it wiser to use new stuff rather than camping rope that has been in and out of mud and such...). Indeed, by loosely wrapping the nylon rope before putting it in the plastic sub-wrap it happily came out tangle-free when it was time to "deflate" the ball.  ( I used a twist-tie to hold the ends closed - which made it easier to undo when stuffing!)

Ellen W suggests using nylon craft cord for the mari core - it's slippery which makes taking it out easy and less tangling....

From Mae A - I do have one hint that may help someone, I use a large straw -like comes with a malted milk-- and cut it in half, put the end of the yarn through it and let a foot or so of the yarn hang out, then start wrapping the mari on the other end of the straw and continue until the mari is as large as you want it, then put it in the plastic bag ( by the way one of the thin bags from the produce dept. in the grocery store is just perfect) then continue as described by Sue H.   The straw does several things, it keeps one from getting the end or the tail of the yarn bogged down, and makes removing very easy and smooth,and makes it easy to re-stuff the ball. Lets clarify that, when the straw is removed and the plastic stretched over the opening of your container there is a nice unobstructed opening to stuff into.   This works for me, hope it helps someone.

From Sue H - It appears to me that the purpose of using a straw is to assure that the yarn/cord/rope tail does not get inadvertently buried; as well as making
sure that as you wind your yarn and thread layers, you will leave enough of an opening to re-inflate the mari with batting when you are done.

If this is the case, then even using a left-over inner tube core spool from some threads will keep the tail in place, and prevent you from wrapping the
subsequent mari layers too tightly around the tail.

To a certain extent, by using the nylon rope that I used the second time, - being of a considerably greater diameter than even rug yarn, insured that my
opening would be large enough to negotiate the refill step.  I can certainly see that if I had used a thinner yarn for the inner core, that using something to maintain the opening space would be useful.

Remember also, using a plastic or even fabric form of liner around the inner core, when gathered together at the tail, should help provide a bit of bulk
around the opening area that you may not necessarily need a "space-saver" like a straw.

A lot will depend on what size container you would be using - hence, what size mari you would be making. For my two bin-temari, I chose those inexpensive "ivy" vases (about 73 - 77 cents each in the local Wal-Mart or Michaels)  Their neck opening is about 3 inches across / or 26 cm exterior circumference - and the body about 38 cm exterior circumference.  My "clothesline" core is about 1/4 inch diameter and I used about 50 feet of clothesline for the inner core.

What you are trying to achieve is using something in the inner core that can be easily pulled out, and refrain from getting tangled or knotted, yet not
leave such a huge opening that it would be too difficult to hide in the end.