If you're talking about engine-based propulsion, it is possible but unlikely.
To give you some perspective, a Boeing 747 holds approximately 60,000 gallons of fuel. Now, the weight of a car is some 3 or 4 orders of magnitude less than that, and the amount of distance needed is similar.
Getting something off the ground at a weight of 2-2.5 tons, however, isn't a trivial task, and even then to continue moving it you need significant airspeed. Airspeed in a nutshell is the lowest required speed of travel required to keep air moving through the engine to provide a given thrust. Most of this thrust in horizontal motion is, if you look through the equations far enough, to prevent the object from falling out of the sky.
I'm not saying this to discourage you, if anything here's a few things to consider when trying to come about a draft of this:
1. Find an engine powerful enough.
2. Calculate the amount of time needed for an overestimate of use (you say you'd only need it for 5 minutes, but what if you can't land right away and you need to stay up for 15?)
3. Calculate a general estimate of the amount of fuel needed for moving a vehicle + engine + fuel (yes, it counts too!) to a minimum altitude and minimum airspeed
4. Apply a general rate of fuel consumption provided altitude and airspeed are met and factor it for the amount of time you plan on flying this thing
5. Find a way to safely land this flying hunk of steel
6. Find a way to condense that into something vehicle-sized that doesn't hinder the other functions of the car (i.e. passenger room, engine power - it can't go 15mph max just because it's towing a large engine!)
Once you have these laid out and you're sure to a decent regard of the results, you have a good conceptual idea. Everything after that would ideally be both necessary and not-so-necessary refinements.
Best of luck. It may seem like a wacko idea, but that's because there have been a lot of "flying car" concepts that have even made it off the drawing board but been gross failures due to any number of reasons. That being said, it's not impossible - people thought the idea of flying was impossible, way back in the day.
Everything achievable is done with an equal amount of time and effort. If you got those, then who knows what you'll be able to achieve.