I really like my 7" Nexus 7. It's Android, and does pretty much all that you mentioned. There is an inexpensive OTG cable that converts the mini usb port to a regular usb port. I've not tried it myself, but you might be able to attach a USB external HD to the Nexus, or use a thumb drive.
There are rumors that a newer, refreshed Nexus 7 2 may be available as soon as July, and at the same prices as the old model, so if you can be patient, it would pay to wait a couple of months.
I always thought I was fairly mainstream, but I guess not when it comes to tablets. I've not been able to find anything that fits what I'm looking for.
* Use: mostly watching video, some book reading, web access nice but not necessary. I want "good" resolution but I don't need hi-def.
* Physical: I need to be able to easily carry it around in one hand or to stick it in something (say, an already pretty full laptop bag) so it doesn't interfere with carrying other things (when, for example, getting on an airplane with laptop and carryon). I think 7 inches may be about right, 10 is probably too big.
* Connectivity: wifi would be nice for occasional web access and to download media files. It can't have cellular connectivity (it needs to go places where they prohibit such devices). No interest at all in buying/renting from online stores or storing anything in a cloud. I will have no connectivity in a lot of the places where I want to use this.
* Storage: lots and lots. I want more capacity than any tablet is going to have. So what I really want is at least 16 GB on board (32 would be better) and a port to allow direct connection of additional storage devices (microsd, USB, etc). I expect to carry a handful of cards or sticks with me.
* Media format: I would like to be able to directly play .iso files. Failing that, .mp4.
* Battery: I'd like to have a solid 8 hours of battery life during video playback. But if it has a charger that can "keep up" with the battery, then 5 hours may be enough.
* Brand/OS: I prefer Android but anything with a track record of reliability will do.
I realize this sounds pretty picky. Really, it's the size, connectivity and storage which are essential. If I can find something which fits those, I'll probably find a way to work with the rest of it. The problem is, so far I haven't found anything that meets them. Everything these days seems centered around the online experience, which is exactly what I don't want.
What is out there for me?