The thing to search for in rental listings, either online or in the paper, is Rooms for Rent. These are typically places you can rent by the week at rents approximately proportional to monthly apartment rents. In general, more likely to be undesirable locations and quality, but more likely to include furniture and utilities.
Put your stuff in storage and crash at friends’ houses. I suggest you go on a trip somewhere, but what are your responsibilities in town and why are they so pressing? Why did you sign the new lease knowing full well you’d be out of a place to stay for >2 weeks? Why don’t you talk to the apartment folks and get them to let you in a bit early? They’re pro-rating your rent anyway.
You might look on Craigslist to see if anyone has a short term sublet in your area.
Extended Stay America is probably the cheapest longer term stay hotel. They’re usually about 250 per week.
I think most/all the leases work the same way in this town. They all seem to end July 31st but don’t open up until mid august. The reason is because the apartment managers need 2 weeks to repaint/clean the apartment for the new occupants. The reason they do it is because this is a college town and 80-90% of the people who sign new leases aren’t in bloomington during the summer anyway so they don’t care about 2 weeks of homelessness.
campin’ I’m a goin’.
In Bloomington, a major university town, you shouldn’t even need Craigslist. If school isn’t in session yet, you should be able to find half a billion short-term sublets. Most leases in town that are anywhere near the university also start on August 1st, so anyone who is stuck with an unrented empty apartment right now ought to be thrilled at the prospect of some extra cash. Haven’t you checked the bulletin boards in the student union?
I know there used to be some landlords who split up big old houses into rental rooms - maybe one of them has a slot? Even if they’re repainting or something, they can only do one room at a time, so they might be able to shuffle you from one room to the next. Or do you have/know a professor who is going on vacation then and needs a house sitter?
Eva Luna, IU student, 1994-95
“Young man, there’s a place you can go…”
does the Y still have cheap rooms?
Get girl/boyfriend? 
I don’t think the ‘Y’ even exists in the form of real buildings anymore. Its all programs and events.
- Inflatable Mattress
- Sleeping Bag
- U-Haul truck ($20/day?)
Be sure to rig it so you don’t get locked in. Drive to a gym for showers.
I spent Mardi Gras weekend in New Orleans on a student budget this way. It was great!
It sure does in my hometown; they just built a new pool building a few years ago. They had rooms as recently as 1987 or so, but I don’t know about now. There are seevral YMCAs in Chicago as well.
$20/day for a uhaul is damn expensive. I can just get a tent for $40 and sleep in the forest instead. That’s probably what i’ll end up doing.
This was definitely my first thought. Or, as Eva Luna sugggests, university towns often have heaps of apartments available for sublet during the summer. Around here, some students end up renting out their apartments for half or three-quarters the price of the rent, because in summer supply is so much greater than demand that you can’t get full rates. Put notices up, or even contact the administrative assistants in the academic departments.
A couple of years ago, my partner and i spent two weeks in Chicago doing research and having a bit of a vacation. We wanted to go in early August, and in early July i sent an email to some of the humanities departments at the University of Chicago asking if any of their grad students had an apartment to sublet. I got a few responses, and we ended up renting a very nice one-bedroom place in Hyde Park, right near the commuter rail stop, for $250 for 2 weeks.
I made a research trip to Sacramento the following summer, and did the same thing with Cal State Sacramento. That time, i actually got a free house for two weeks, all to myself, in return for looking after the plants and the pets while the owners were on vacation.
Are there any youth hostels in Bloomington or nearby? You could look into that. Those are usually pretty inexpensive.
I’d go camping. However, I also wonder what your current landlord is going to do with your room for 2 1/2 weeks. If s/he is expecting a new student, that tenant will probably not arrive on August 1. What about offering to pay for your current room until you move out, say 5% more than 18/30ths of your current monthly rental just to make it attractive?
That’s the perfect time to go camping. Just search on Google maps or Google - “bloomington indiana camping” and you will find several good options.
McCormick’s Creek State Park
http://www.mccormickscreekstatepark.com/
Monroe Lake (a state-operated park)
http://www.in.gov/dnr/parklake/properties/res_monroe.html
Cloverdale Campground
http://www.hikercentral.com/campgrounds/104790.html
Don’t waste your money on an expensive hotel room! Enjoy the great outdoors. You won’t regret it! 
Good luck.