We spent 9 days in Dubrovnik in March 2011 and loved it - could definitely imagine a month (although we did have a car which is likely influencing my opinion).

There are lots of things to do nearby and a good (I think) local bus/ferry service so you could do short overnight trips as well.

We stayed in a terrific apartment with a stunning view of the Old City - a bit of a climb up but nothing dramatic and there are lots of buses. I considered staying in the Old City but was happy we chose the view as I never tired of it. The family who rents these apartments is lovely (and onsite if you need assistance) and they offer excellent rates for off season

http://www.dubrovnik-accommodation.net/index.html

And here are my sunset/sunrise time lapse videos - I was having fun with a new iPhone time lapse app!

http://www.youtube.com/watch?v=PVgdtx7a71w

http://www.youtube.com/watch?v=unlMsbHQmrI

I have visited Dubrovnik several times...it is lovely, but one could get restless spending so much time
In a small town( and out of season) in a country whose language one does not speak.
The weather can be cool and sunny, but the days are shorter...
Except for Split, the costal towns are even smaller with less to do.
I am not sure about bus/ ferry service out of season.

We spent 4 days in Dubrovnik in November 2010. We had one warm sunny day, one fair day and 2 days of torrential rain and thunder storms which resulted in significant flooding in the old town. The local people working at the hotel told us this is normal at that time of year. the old town area of Dubrovnik is extremely picturesque but is quite small with limited things to do and see for more than a few days. Up to you but I would not personally be excited about spending a month there, especially at that time of year.

Reading the other comments perhaps I should elaborate that we usually travel off season and spend extended periods in rental accommodation so as to experience 'living' somewhere. Therefore we are used to weird weather patterns and fewer tourism amenities due to it being off season. Hence my enthusiasm for a month in Dubrovnik off season, but it might not be everyone's cup of tea!

I think the issue is really one of staying in what is a tourist economy when the tourists aren't there. I live in such a place and love it, but not everyone does. We all talk about loving to see 'em come and loving to see 'em go, but when they are gone, many restaurants and shops close, nightlife more or less disappears, and the streets that seemed objectionably packed now seem to many to be sadly empty. Houses are dark, and residents take up their own social life, from which tourists and seasonal residents are generally excluded.

If you like solitary hobbies -- birding, photography,reading, painting or drawing, making music, studying architecture, for example -- you are likely to be quite happy. If you want to meet people, going to church or taking cooking or language classes or volunteering to help people with their English will open some doors or you could do a volunteer project with a local group if you have or could develop language skills.

Can you deal with interruptions in travel? Dubrovnik gets the bura in the winter and flying is difficult to non-existent at times. Mountain roads may be closed by snow, ferries cancelled by wind. There are lots of people who write here -- and I don't suggest that you are among them -- who depend on having fairly tight schedules of "things to do" to be happy.

It is always funny at the airport or ferry terminal here to here Big City folks saying, "But you don't understand, I HAVE to be back in Big City for a meeting tomorrow!!!" And the airline or ferry folks play on their anxiety by saying, "Well, you know that at this time of year, sometimes we can't run for two or three days at a time!" And sometimes they can't.

I can't say whether all these things sound like obstacles to you or whether they sound absolutely wonderful, but you need to consider them.

Everyone, thank you very much for your feedback. I still want to go to Dubrovnik, but I'm rethinking the amount of time I'll spend there, but am hoping for at least 10 days to two weeks.

Dubrovnik, being so far south, could be hard to go elsewhere unless you were working furher south. A mid point in Croatia might be better such as SPlit or Zadar, but not sure there are is enough there.

Both Split and Dub have decent aiports to travel elsewhere. I found Croatia airlines in country to be very cheap and easy to get around. Some flight do not have to go through Zagreb.

If it were the summer, the ferries would be running and that would be nice but they have limited runs in winter.

I too travel in November and was there last year and the weather was great, especially the further south you went.

I've spent some time on extended travel Croatia, including in the off-season. I love Croatia and think you can spend a month anywhere and get as much out of the experience as you put in, but... personally I would divide the month into maybe two weeks in Dubrovnik and two weeks in Split. Allows for a really relaxed time still but more options on daytrips. Also, even though you'll get off-season discounts, Dubrovnik is still one of the more expensive places to be in Croatia.

From Dubrovnik a couple of bus day-trips are possible such as Herceg Novi, Montenegro which is like an hour away and costs like €10 or even further into Kotor. Or to Mostar which is a couple of hours each way but also cheap (will be colder inland so dress for it).

From Split, easy to go to Trogir for dinner or lunch or for the day, it's just on the other side of Split airport. Easy to take the ferry or catamaran to the islands (Korčula, Vela Luka, Hvar) for the day, as these regularly, even in the off season. We even caught a movie at Joker mall in Split, they show a lot of English language movies there with Croatian subtitles.

We had great weather for all of November last year, with the exception of a few rain days, one with the cold 'bura' wind. You'll probably be there during olive harvest. Bet you could find some locals that could use your help We did last year. Olive picking is hard work but was a great experience. Good luck!