I would like to build my own network storage device by programming it
in Linux.  Ideally the device would be inexpensive, small, have one
IDE drive and a network port.  Internally, it would have flash memory
into which I could flash my own firmware.

The device would be somewhat analogous to what SVEASOFT and OpenWRT
have done with Linksys and other routers wherein one can flash custom
firmware and do all sorts of fun things.

Does such a device exist or could it be reasonably built?

you can try freenas http://www.freenas.org/

http://www.hak5.org/wiki/Episode_2x02
shows you how to install it and run it in this episode, I believe they
also mention some alternatives as well. good luck.

I'd take a look at either the Soekris products ( www.soekris.com ) or
the Linksys NSLU2 ( http://www.nslu2-linux.org/ ).  Neither one is
exactly what you are asking for but both offer quite a lot of
possibilities for the money.

I'd try searching around eBay for a good deal on either.