In reality, you can build a rocket. But for a spaceship for space travel, you need much more than household items.
There are two things we must consider here, intelligence level and ship capability.
We seem to be working off 'ordinary human' and 'standard NASA rocket' here, in which case the simple answer is 'No, there's no way Keen can do this.'
However if we assume that Keen is indeed a hyperintelligent 8-year old, then we can ask 'Is it physically possible to make a spaceship out of household components given unlimited intelligence'?
Now I'm going to ignore all physically impossible things, faster than light travel say, since if we don't we don't have any logical basis and can just go 'Keen invented the Jobs field, which allows you to make stuff out of junk with no effort.'
The first issue is of course, thrust; this is a big issue, perhaps the most important issue. Keen will need to leave earth at an escape velocity of 11m/s
http://en.wikipedia.org/wiki/Escape_velocity to leave the earth. Or at the least he will need to provide him and his rocket with enough kinetic energy to overcome earth's gravitational pull (And air friction). This is much more than the energy in a single bottle of alcohol and a car battery.
http://en.wikipedia.org/wiki/Rocket_propellant
Ergo Keen must be obtaining energy from a nonchemical source. The flames from his ship suggest that it moves by emitting exceedingly hot plasma (Normal oxygen-fuel flames don't work in space, and carrying sufficient chemical fuel to fly across the galaxy using them is unfeasible, ergo these cannot be fuel-oxygen flames.) This suggests a canister of compressed fuel, solid xenon perhaps, or maybe gas collected in situ, that is then ejected via ion propulsion at high speeds and temperatures.
http://en.wikipedia.org/wiki/Ion_thruster
The orange color of the flame suggests sodium impurity, as would be encountered in using impure materials as the fuel source. This suggests Keen moves his craft by vaporizing mass that he picks up wherever he is, meaning as long as he does not spend too long in space, he will always have propulsion fuel.
But how to ionize that fuel? It can't be via the battery, we need a massive, definitely nuclear source if we wish to keep Keen's rocket's weight down. A small and efficient hydrogen fusion reactor would suffice, if strong enough magnets could be produced to keep the plasma volume down. Everclear would make a fine hydrogen source. The only setback is the reactor would not be able to run full time; an idling reactor would burn through a massive amount of fuel, so it would need to operate in bursts, and thus have some way of starting it from scratch. A car battery would do.
The next question is that of airtightness; even a small leak would rapidly deplete the air in Keen's small craft, so it is vital that this be so. Fortunately the smaller a craft the easier this job is, and there is no reason multiple layers f soup cans welded into a shell wouldn't serve the purpose. (The problem of heating on reentry can be treated here, it only exists if you wish to freefall to earth, which also gives you the problem of ahem, landing. If Keen uses his propulsion system to slow him down, cans would suffice as shielding.)
Providing Keen doesn't do anything too strenuous, or hit space junk, a plexiglass cockpit and rubber seals should work fine for sealing as well.
The problem of control is trivial, Keen's ship seems to operate merely as a conveyance from place to place, and so aside from simple controls governing the stop-and-start of propellant, a simple joystick interface would suffice for his purposes.
What you really need, not what you think you ought to want.
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or get the mythbusters guys to do it.
