temperature light came on after only driving a short distance

So job had homemade speed bumps that were too high and would scrap my car everyday. Out of nowhere one day I start my car and go about 500 yards and the engine temperature light came on. Car was up to 250. I drive a bus so where I work is all mechanics. Told him and he said I needed a new radiator. I did not have the money for it so I kept coolant in. Drove it two weeks on just refilling coolant. One day I drove and it went up to 300 and then cut off. Smoked not a lot but enough. Mechanic says I need a new engine. What exactly is wrong what and what all needs to be done? I do not have a mechanic who does the job but not quite right. And it is an aluminum engine.