Hi, it is theoretically possible to move into a rental in 5 days, it is tight, cause the property manager has to check your credit etc. If you are buying a house with cash, and it is a vacant property/regular sale, it may be possible to close within 5 days, but it is very tight.
For a renter it could be possible, since we are at the end of the month when most move out move in happens, but finding the place to move in to will be your biggest challenge, you best get going on finding a place.
If your looking for a rental, it is possible to move in 5 days but just depends on where your looking and how much your wiling to spend. In regards to purchasing a home it is also possible ONLY if you are paying all cash and only if it is a traditional sale and is vacant. Short Sales and Bank Owned Properties (REO's) cannot close in 5 days. Now if the home for sale is a FLIP property, this the home that can close quickly as they are vacant and usually fixed up to the point that repairs are minimal. Hopefully this helps.