There is no bike that exists that can do all of those things equally as well. The closest thing is a cyclocross bike but you won't find one for $300. It is like asking for a car that only costs $3,000 new that can haul a horse trailer, race on the weekends, and ride over sand dunes in the desert. It just plain doesn't exist.

You are asking for a cross between these things AND for it to be very cheap. It is unreasonable.