New acquisition -- Score! Oh wait, what do I do about THIS? Help a novice, please.  

Vodder pieces are on a framed base with legs, so nobody would have literally cut the legs off.  The screws would have been taken out and the base removed.  The plinth base could be someone's solution for a missing base that got lost during shipping, etc.  Or, the base got broken, and the owner did not have the skills to fix it, and the plinth was their solution.  Or, they just thought a plinth would look better back in the 70's, and removed the base and threw it away.
Finding a Vodder base for the piece would take a lot of luck.  It is possible to fabricate one, but it would be quite expensive. 

Thank you.  Any ideas as to what would be the best thing to do?

If you could get four teak legs or some cylindrical stainless steel legs of the appropriate size, you could use either blind nuts or mounting plates like these to mount the legs directly to the bottom of the cabinet.

Arrange the drawers to that they make a V. That is why the craftsman put it there. 
 
The case is veneer. Don't sand through it. The drawer front _might_ be solid teak. Usually they would use a solid piece only on the top where the recessed handle needed to be cut away, and then put a secondary wood or plywood below. But this appears to be solid teak. Or it could be a secondary wood like afrormosia or mahogany that would look dark like it does in the photos you have posted. 

Thanks, tchp, yes, I like the idea of legs, that is what I'm going to try to do. I discovered the salvaged furniture parts section of eBay and will hope for a decent match. Maybe once the plinths are removed I'll be able to see where the original base and legs were attached. 
Thanks too, Leif E. Yes, I was aware of the pencil V thing and used it to figure out the drawers on the smaller piece, just wasn't sure how the V on the big one was meant to go. But it's put back correctly now, thank you. 
In the daylight I can now see that the case is veneered just as you stated -- it's pretty thick veneer, compared to other things I've seen. So can I gently use fine steel wool and teak oil to clean this safely? Or another method?
I very much appreciate your advice.

Whatever you do, please don't put hairpin legs on it.

Well, the good news is nothing I'd do could look worse than they look now, so the bar's really low!
But seriously, I was thinking to try to get plain wood legs in a similar shape to the original ones, and hope they will more or less go away, and I'll only see the lovely drawers when I walk into the bedroom. 
What do you folks do when you see things like this, that are half lovely and amazing, and half depressing and ugly? Just curious.

Hi.
 
What do you folks do when you see things like this, that are half lovely and amazing, and half depressing and ugly?    
 
It would depend on which half (top or bottom) is depressing, and his credit card limit.
 
Best,
 
Aunt Mark

Ha!
You know I think I just described much of life itself.

A quick, temporary solution might be a set of IKEA Capita legs.  They do not cost much, and there were quite a few Danish cabinets that came with both a wooden legged version, and a metal legged version. 
http://www.ikea.com/us/en/catalog/products/20049538/
http://www.greatdanefurniture.com/Danish-Furniture/VodderSideboard26-22m...
 

There is a possiblity your cabinet originally had teak and metal legs as an available option, in addition to wooden legs.  You should be able to tell what it had originally if you remove the plinth, or look at the bottom of the cabinet.  This might be the easiest, most authentic solution for you to have fabricated.
https://aleph-01.kb.dk/F/BDP3URD8F9GHFKM6BTI1PEQIS2E197R711N5DCNTCTGTDHI...
 
 
 

I forgot these chests came with metal legs with teak boots.  That would look sharp!

Wow, thank you both -- that's an interesting and tempting option!  I now am looking forward to seeing what the bottoms look like. Great -- I am feeling even better about this purchase thanks to you all.

Can someone explain the V thing?
I don't get it.I actually have this cabinet. Never knew about a V on the back do the drawers aligned. Please help with photos. Feeling dense.