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Without thinking much about it, my first guess would be the Meatpacking District and Soho stores in NYC.

They're just over a mile from each other. You could walk from one to the other in less than a half hour, 15 minutes if you're quite fit (5 minutes in a cab).
 
The Grove Dr. and Beverly Blvd. stores in LA are just a fraction of a mile further than the two NYC stores I mentioned.

The two Plato Alto stores are actually the closest I've found - 1.2 miles.
 
Apple Store Northridge and Apple Store Topanga are pretty close.

6.5 Miles

These are both pretty big stores too. I'd say they are larger than the one in the Grove.
 
I live just northwest of Denver, where there are actually 2 Apple stores within about 8 miles of each other (in addition to 3 others further south). Denver has nowhere near as large or dense a population as NY or LA, but the Flatiron Crossing store near me is in an upscale mall near a high tech business park that houses Sun Microsystems and Level(3), among other companies. And the Twenty Ninth Street store, just 8 miles up the road, serves the University of Colorado crowd and the affluent residents of Boulder.

Each location makes sense in terms of the shopping environment, and each serves what Apple seems to consider a favorable demographic. So I can easily see stores just 1 mile apart in places like Palo Alto, etc. I'm actually a little surprised that there aren't two stores even closer to each other somewhere.
 
I found this tread when I Googled for the closed Apple store to each other. Sydney is soon to get a new Apple store quite close to the first Australian Apple store in the city and I wondered if they will be the closest. Info on new Broadway Store.
The two store will be 2km (1.2 miles) apart, straight line distance, further by road. Similar to the New York City distance.

I knew this thread has been dead for 3 year, I thought it may interesting to see what that current closest stores are today, Apple has opened many stores in the past 3 years !!!
 
Here in Vegas we have an Apple Store in the Ceasar's Palace Forum Shops and another in the Fashion Show Mall just north of it. If you were to draw a straight line between them I think it would be under a mile technically.
 
Yup, the two Palo Alto stores are very close to each other.

I've jogged from my apartment which is a few blocks from the downtown Palo Alto to the one at Stanford Mall. It isn't more than a few miles, if that.
 
I live in Colorado and can easily bike to the aspen grove store in maybe 20 minutes. The next closest is about 8 miles away so I am limited to apple stores. I do have to drive if I want to go to the other one in Denver.
 
The two flagship stores in the centre of London (Regent Street and Covent Garden) are also close together.
Screen Shot 2016-10-05 at 07.48.38.png
 
These two are in the same city, on the same street, and are actually 150ft away from each other. The Glendale Galleria Apple Store is actually the first Apple Store ever opened (debated a lot).
Screen Shot 2017-01-02 at 2.28.41 PM.png
 
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