Polishing oxidized eames 670 bases-any recommendations?  

I would think sandpaper, no matter how fine might
be a problem. I have polished oxidized aluminum with a cotton buffing wheel and a graphite stick passed over the spinning cotton wheel and then brought this to bear on the surface and achieved a mirror like finish.
There is someone on this forum who I believe has a great deal of experience with metal. His forum name escapes me at present, but I think he is based in Scotland and designs and works in metal to great effect.

No sandpaper - polish the...
No sandpaper - polish the base using a cotton mop on a spinning wheel and with a suitable polishing soap for aluminium. A proper polishing machine (basically a bench grinder fitted with a mop) will give better results, and much quicker, than hand polishing so well worth buying and they've never been cheaper (thank you China)
The link is for a typical machine - there's 100s on the market
http://www.diytools.co.uk/diy/Main/sp-2-7075-6468-proxxon-28030-grinding...

polishing aluminum
can be very satisfying. All metals have their tricks.
It takes a bit of experience and confidence. And following the steps.
Removing the oxidation is very important.
No real quicky solutions but just seeing it being done in a few
examples will give you some ideas about how to go about it.
I have a vintage airstream so that is my experience. I have a file of
research. Seeing it done in such a large scale will make a chair base
seem like shelling a peanut.
Nuvite is a popular polish. Also used on aircraft.
Maybe not very helpful for your specific problem, sorry, but getting
familiar with a few google searches and knowing that it is possible i
find very helpful to tackling any restoration project.
http://www.wikihow.com/Polish-an-Aluminum-Trailer
http://www.vintagetrailersupply.com/Nuvite_s/19.htm

is it just me?
I find watching these videos fascinating.
This does show what is possible.
And how great for an elder pro to show his skill and knowledge.
My Airstream is such a disaster of dull. Watching some of these videos
gave me hope and doing a very labor intensive test area i am now confident.
I just need a week off to complete the project...
http://www.youtube.com/watch?v=Vp-aL7ega5o

I should add
BTM and Paulanna are correct in the advice about using a cotton buffing pad
with an alum polishing compound. And a good buffing tool is not that expensive
and will come in handy for many projects.
(I threw in some long-winded examples for a yuk. Just thought a few visuals
may help)

Powder Coating
Some Eames aluminium has a clear powder coating on it. You will not be able to polish this without removing the powder coating, if yours is similarly coated

Believe me,this is SO oxidized
I don't have to worry about powder coating,as whatever dulled the surface is going to be ruubed away! Thanks fellow forumers,for all the helpful tips.Off to buy me a Dremel with a cotton polishing attachment.

Nix the dremel, go with a circular polisher
I don't think I would have the patience to use a dremel to polish an Eames base. Assuming you're talking about the hand-held models, you'll be going through a lot of cotton attachments if your base is that oxidized.
I would get a circular polisher (not orbital) and use aluminum mag polish...that will get it shining in no time. I've got a Dewalt 849 that works wonders. The Makita 9227 is another good one. A circular polisher can be bought for about $180 new or about $60-100 used on eBay or CL.
fm

Doing it with a Dremel is a...
Doing it with a Dremel is a world of pain you don't wannna go to. The suggestion of buying a bench mounted polisher is in itself a compromise when the very best tool is a floor mounted polisher that you stand in front of. Large areas done with a Dremel tend to look streaky anyway.

a dremel is best for 'smalls'
like a watch.
The pros here tend to forget about the DIY attempt. A good polishing kit is
available to fit a household drill. Would work well on a small flat plane of a
670. Not everyone has even a garage. Some here are in apartments and most
have a household drill.
I buffed my base with 'astonish' and it looks great. Soft cloth. Not mint but fine.
A hand rub.
And i have a 5 thousand sq ft shop. Every tool in the book.
DIY advice needs to be simplified sometimes.
I have 50 yr old cork floors but actually like the patina. No quick fix to make them
look 'new'. Replace? not a option. Beautiful as they are.
A 670 base does not need a mirror finish but it is possible.
Such a small surface area really can be done by hand. A passionate hand and some time.
But a household drill is a homeowners tool owned by many.
This eye-crossing advice is confusing to a DIY. Who wants a stand-up buffer
in their home?
http://www.astonishcleaner.com/?leadsource=PS105&s_kwcid=astonish|1589129749&gclid=CPTXj7ai-JQCFQwuHgodswdgrA

A 670 is a valuable and...
A 670 is a valuable and desirable chair. When it comes to maintenance and repair I don't really see how 'second best' solutions are appropriate. If you can't do a job properly at home because of lack of tools, space or skill then you should pay the money and take it to someone who can. You don't see pros using powerdrills as polishers because they don't do the job properly - and heaven help you if the bolt for the arbor (necessary on these) comes into contact with your base....

this kit
uses a power drill.
I prefer Nuvite polish. Used on aircraft and airstreams.
http://www.caswellplating.com/buffs/alpolkit.htm

.
Am I alone?I like a nice over all grey...not all this glitz...shine has allure but vintage surfaces are often beautiful...

Chrome sucks
But aluminum is cool... polished, anodized, or tarnished.