Definitely, being in space is not required. (But if it is in space, then I want ships that follow all the rules. I played a video game recently that was a first-person shooter set in orbit -- not much gravity to speak of, but your suit had its sensors tied in to speakers, so you could still "hear" things. But if you wanted to be quiet and stealthy, you could shut down most of your suit's systems, and if you did that the game's HUD went away and so did the sound. Neat hard SF-ish twist.)
The science and the accuracy are what make it "hard SF", to me -- if there's a lot of handwaving or flatly impossible things going on, it's not hard SF. With regular SF, I like stories that take one 'unreal' thing (time travel, FTL, telepathy), and then extrapolate a world or story from that, without adding lots of other unreality. With hard SF, I like it if the 'unreal' is something that's technically still possible, like a big space program that in real life would never get funding, but would be physically possible to build if someone had the money.
"Exhalation" is an interesting example. Lots of not-real tech in there (mechanical AI? we don't even have electronic AI yet), but nothing that we know for sure is not possible. The great thing about "Exhalation" is how the exposition really drives the story, plus it brings up lots of interesting philosophical considerations, and it has some drama (if sort of slow-moving drama). If you could find more stories that do all those things, that'd be great.