I have always wanted to go to an airport and just ask which flights are leaving soon and have availability, then just go.
You can find somewhere to stay while waiting in the terminal, and in my experience you can get an absolute steal if you use a site like booking.com for a same-day check-in. Crawl google maps for a while and good places to eat and things to do will come to light.
A visa could limit this; Europe is fine, places like the USA and Canada require pre-entry details that take a couple of days (though I have this for the USA already).