Is it possible to build a turbojet engine with aluminum cans? (Soft drink cans)
No, you cannot make an entire model jet engine out of aluminium.
Most fuels burn at 1000+C and aluminium melts at around 660C: Even if you keep the burning contained in airflow (thus away from the walls) the heat transfer will still damage the aluminium.

You could make the majority of the engine from it, though: Just as long as you find something more suitable for the 'hot' parts.