Where should one look for short term residency options - less than 6 months? Sublets? Hotels?

Is there a college housing office nearby, go there and ask for contacts.
There are a number of hotels that specialize in the short-term stay market.
You can also find a hotel near where you want to stay and walkin request to discuss with the manager and propose what you are willing to pay.

I have had occasions over the years to need temporary residency for my family and I. Through trials I learned that residence inns met our needs in terms of comfort and cost. My stays would vary from 30 days to 5 months or so. In todays world there are many brands to choose from and finding what fits ones needs is pretty easy. My experiences pre-date laptop computers but one computing and tele-computing became a method of getting some work done while on the road it became important to me to find free wi-fi. Seems most of these extended stay facilities offer a variety of discounts. One catch to the discounts, you must ask - they usually won't voluntarily lower the price because you are a nice guy. Do military facilities have transient barracks conducive to family living? The biggest concern is economic and kitchenettes are a must. If one is aware of this need for temporary extended stay will ahead of time, that is the time to look for prices and make reservations. That is my 2 cents worth.

Couple good answers here and one I didn't see. Many small local "mom and pop" motels, some with chain affiliation, have one or two efficiency units. These places are generally cheaper than the big residency and extended stay hotels and are willing to give decent discounts to military families.