Most homes that HUD or VA have a 10 day offer window. REO property can be another matter and an offer may be accepted as soon as it's available but it will not be a low offer.
first I would like to know: are you a realtor? because if you are then I can understand your question. To answer your question, it is possible that the offer was presented some time ago and the prospective buyer was already preparing to close on the home just waiting for the final acceptance.
to answer question not every home is listed in MLS, exclusive listings can be part of a particular brokers inventory. Yes, a home can close in a week, while not common at all, if the buyer is purchasing the property all cash ( without financing) provided that all participants can perform the due diligence and deliver a clear title withn that time frame.