set of Jalk tables in need of love.  

DHutch 17,
Sorry to hear that...Are you certain the end table is by Grete Jalk? It should be solid wood, not veneered, and therefore easier to refinish.
Your photos are not working, try posting them again.

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Pictures
Solange - they are for sure Grete Jalk. The upturned lips and legs are solid, but the tops are veneered.
Hopefully the pictures work this time. Whatever I do to them next, it for sure will not be paint 🙂

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Ouch! the worn through veneer is going to be very tricky to deal with, perhaps staining black is one of very few viable options.
Great tips Mark, although how long does a drinking glass remain in one place in your household, I should imagine it would spend too much time in your hand to be of any use as a decoy.

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Interesting DHutch17,
and a little disappointing. I thought the end tables were solid teak. My table is not perfectly bookmatched and the ends are exposed, so if it's been veneered to plywood they did an excellent job of concealing it, as I recall seeing no obvious sign. Mine has the Glostrup label and looks just like yours...darn...your photo is convincing testimony that it is not solid teak.I will look at mine again next time I'm at my country house.
I guess I should consider myself lucky I didn't damage it as I went to town sanding, believing it to be solid.
Mark's suggestion is a good one, and would no doubt look very attractive, but I suspect you wouldn't be entirely happy with it that way, and what's that saying..Once you go black, you can't go back?

Ebonised tops
with teak legs would be my way forward I think!
Hello Mark x)

When we say stain black...
When we say staining black, are we talking ebonizing?
Mark - I like the suggestion of black. That would cover the damage best I think. I love the warm red tones of the teak though... hmmmm. Has anyone ever had teak veneer replaced and know what that might run me?
Thank you all for the help!

Holiday hugs, canned hams, and highballs to you Robert!
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re-veneering
You could try re-veneering yourself.
How to: http://www.rockler.com/articles/veneering-with-contact-cement.cfm
What with: http://www.woodcraft.com/product/2008681/18257/teak-veneer-4-x-8--10-mil...
The lip on the edge of the table will be tricky to do. You could run the veneer all the way up the lip, and trim it there. Or you could run it part way up and then sand the edge of the veneer down to nothing. There will be a seam where the solid lip meets the veneered top, and you could preserve this seam by cutting the veneer and applying another piece over the lip. Or not worry about it.
I have also seen sanded through veneer that has been painted with faux-grain that was virtually impossible to see.
Sorry to see they are so damaged under the paint!

From the underside of the...
From the underside of the table with a very sturdy razor kinfe, working along the seam seperate the solid edges free of the table top. They are usually just glued on. A heat gun makes this a little bit easier. You can then sand the old veneer off (think power sander) down to the substrate, reveneer and reapply the edgeing.

for a newbie
Can I realistically reveneer these tables myself? I am not terribly skilled when it comes to woodworking, but usually read up and give it a go... how else would I even learn, right?
I guess I can do anything worse to them. I like the idea of ebonizing the tops as well but I still have one end table that is completely flawless that I would not touch, and thus it would not match.

give it to a local woodworker
don't go mental on it yourself, these pieces
are to nice for a home project.
(in my opinion)