I’m a billionaire and an aviation nut, as well as a hippie.
Could a real Jefferson Airplane, part antique trimotor triplane, part San Francisco Victorian house just like in the drawing, be built that would fly? Side-door entrance, wire wheel landing gear, upper story bay windows, grow room in the rear. Say the word and I’ll start tomorrow!
It could be groovy.
I’d be happy to assist in any way I can, right after I’m done bathing at Baxter’s.
I suspect I would try a scale model before dropping acid, jumping in and heading for the sky.
In my inexpert opinion, having only flown model aeroplanes, that thing won’t get off the ground without jet assist and would be near uncontrollable approximately 2 seconds later.
However, it would be really cool if it could.
Given an engine powerful enough, one can fly a barn door.
Brian
Count Gianni Caproni beat you to the punch:
Stranger
Outside of the frills and nonsense, it is a standard layout, enough wing to carry it and long enough moment arms for balance. It would be a straight forward RC design.
The easiest way to accomplish this is with a hot air balloon approximating the size and shape of the plane imagined in the picture.
Paging Howard Hughes
I think the wings would have to be significantly larger, and the “house” parts would have to be made of lightweight materials and the walls would be much thinner than those of a real San Francisco Victorian house, but I think it would be possible. A couple more engines might be necessary as well.
The old joke about the F4 Phantom was that it was proof that you could make a brick fly if you gave it big enough engines.
Here are some pictures of hot air balloons made to resemble airplanes, and some other shapes. I know it’s not the same experience as flying a functional trimotor triplane around the country while tripping on some mellow acid but it’s more feasible and still cool looking.
Taking it to the next level -
With enough thrust, ANYTHING can leave fly/leave orbit.
Slightly more seriously, given the power to weight ratio available on modern engines, AND the ability to make strong and incredibly light composite materials/carbon fiber, it’s likely feasible even if it would be horrifically unstable and inefficient. At least, if you’re just aping the outside appearance and not furnishing the “house” portion. 
To achieve flight the object must balance 4 components:
weight, lift, thrust, and drag.
Flying and leaving orbit are two different things.
There’s a (slight) possibility that the first object to leave the Earth’s orbit was a manhole cover that was not properly secured during an underground nuclear weapon test. Some folks have analyzed the blast footage and calculated that the manhole cover had enough velocity to escape the Earth’s gravity. It should be noted however that others who have analyzed the same footage claim that it wasn’t going that fast.
Either way, I wouldn’t exactly say that this manhole cover was “flying”.
A nuclear explosion does however provide a rather whopping amount of thrust.
Well, yeah, I was thinking of various versions of Orion-powered “flight”. 
Even if it did have enough momentum to punch through the atmosphere and theoretically reach orbital altitudes, the heating from ram pressure (compressing the air in front of it) would have almost certainly reduced it to molten droplets long before it left the atmosphere. Rockets are profiled to minimize such drag, and they actually limit thrust until they rise high enough above the thickest part of the atmosphere such that the heating doesn’t threaten the vehicle structure or payload without an excessive amount of thermal protection (TPS). An impulse-launched manhole, on the other hand, has maximum velocity from the beginning of its flight and will rapidly heat up to melting temperature in the first few thousand feet of ‘flight’.
Stranger
It’s a sign of the times that I read that as Onion.
Okay. Couple of thoughts. Balloons shaped like planes won’t cut it. I want a real airplane that flies.
I would like to see a scale model though. A flying one, not big enough to ride in. We’ll save the full-size one for after the Proof of Concept.
And that’s how Jefferson’s Airplane became Jefferson’s Starship.
@frodo gets it!!!
My work here is done.
