Is this hopeless?  

Wow, that's
amazing, tchp! I have seen Barkeep's Friend around for years but never knew it had oxalic acid in it. I'm going to get some even if I don't have any dark rings right now to deal with.

tchp, your pix are worth...
tchp, your pix are worth 5000 words. Very impressive. Was the mold stain very engrained or was it mostly on the surface? Did you use the liquid or powder Bar Keepers Friend? Thanks!!

It is the powder cleanser. ...
It is the powder cleanser. Not the one for Stainless steel.
Black stains on wood are typically an iron oxide stain, made from minerals in water. Oxalic acid dissolves these kinds of stains.

Follow up instructions
Thcp can you give us some instructions on how to use the Bar keepers friend to remove the rings. Thanks for all of the help!

Sorry, been out of town. ...
Sorry, been out of town. Just make a thick paste with water and the cleanser. Not sure if using distilled water would be better, but it could be advisable so as to not introduce more minerals into the wood.
It takes numerous applications for heavy stains. Apply the paste, allow to dry, then wipe off and re-apply. If the stain shows fading after a couple applications, there is a good chance it can be removed completely.

Barkeepers Friend
I wasn't aware that there was a powdered version that is 'not for stainless'....can you clarify?

All four of their products are
recommended for stainless steel (link below). I think the one with the gold label is the cleanser that has been around the longest, though.
http://www.barkeepersfriend.com/premium-household-cleaning-products

On a small tangent . . .
if you buy some Bar Keepers Friend, try it out on any porcelain cast-iron sinks or tubs you have around the house. It will get it so clean and white, you'll be kicking yourself for not trying it sooner.

Yes, I should have clarified...
Yes, I should have clarified that. I do not think their specialty cleanser for stainless steel has oxalic acid in it, but I am not entirely sure about that. I use the gold can, which has a notation on the front that says "contains oxalic acid".
You can buy oxalic acid at woodworking supply places, which is sometimes sold as "wood bleach". It is a stronger concentration so you might need to work out how much to dilute it in order to avoid bleaching the natural color of the wood. I have never used straight oxalic acid myself.

As cdsilva said, Barkeeper's ...
As cdsilva said, Barkeeper's Friend is excellent for cleaning porcelain sinks, especially removing the gray metal pot marks that porcelain sinks end up with that are otherwise very hard to remove.

Update
I tried Bar Keepers Friend on a small area on back of the back panel. I made the paste and scrubbed it with a fine toothbrush. After the paste dried up, I'd wipe it down with a damp cloth. I repeated that 3 times.
It sure took care of the stain but it also took off a bit of the finish too. Did I scrub too hard? Or it's inevitable? Should I just apply the paste with a cloth and just slightly wipe in the affected area? I'd like to know before moving on to the more affected area. Thanks.

There's a lesson here, kin.
Always test on small, inconspicuous areas before committing to any unknown product or technique.
Also, advice offered on free and anonymous internet message boards is often worth less than it cost.
Really sorry about your chair.

Indeed, tktoo. I remember...
Indeed, tktoo. I remember someone saying something to that effect on this forum before. That could have been you in fact. I was rather cautious when I tried it. That's why I chose the back of the back panel. The area maybe around the size of a quarter. Anyway, will proceed with even more caution now.....

.
Why don't you try not scrubbing this time. Tchp said to wipe it off. Barkeepers friend might take the finish off through chemical means. Or it might not. But I guarantee you will take the finish off with scrubbing. And the oxalic acid doesn't magically work better courtesy of scrubbing.

It seems like this "barkeep" product
is intended for use on hard surfaces and was recommended simply because it happens to contain 5% of an active ingredient commonly used to remove iron stains from certain types of wood.
Oxalic acid is cheap and easily obtained if and when it is called for. I'm all for DIY and using what's on hand, but I would never advise putting an abrasive powdered cleanser on finished furniture.
Generally, it's a good idea to start with the least aggressive and potentially harmful products and work incrementally up from there. Oxalic acid would be among the last resorts if the cause of the stain is unknown.
Usually, I start with a clean rag dampened with water. I recommended trying a test with diluted household chlorine bleach on a cotton swab because I've had good results removing mildew stains with this technique in the past and it won't harm the spray finish HM put on later DCMs.