Can you become and Eagle Scout in four years?

Yes.

You can work on Tenderfoot, Second Class and First Class requirements at the same time and there is no requirement for a specific period of time served. A common goal is to have new scouts earn First Class within a year but technically you could do it within weeks.

Star scout then requires at least 4 months, Life Scout 6 months and and Eagle scout 6 months. So it is highly unlikely to actually occur, but it is technically possible to earn Eagle within a year and a half.