CONGRATS!!!!! I lived in the Arcata/Eureka area for 6 years, but that was in the 90's.
It's a gorgeous area, and the hospital was very nice. (I only saw the ER and lobby.)
There are a ton of apartment areas around, and a lot of houses have been transformed into apartments, too, which is always a nice way to go. There are lots of older, gorgeous homes that are now apartments.
You can also look in the town of Arcata if you don't see anything you like in Eureka. It's only 7-10 miles from Eureka, and traffic shouldn't be too bad during "nursing hours". It's a college town, though, so be careful of anything really near the University. Noise, noise, noise.
You can start by looking right around the hospital. Wouldn't it be nice to just walk there? Also, an idea would be to call the local police department and ask them if there are any areas you should avoid. It's been 15 years since I was there. lol
Again, congrats. I'm SO very happy for you!!!!! Let me know if you have any questions. Feel free to PM me anytime.