designing an amplifier with specific characteristics
To get a gain of 50 with a collector load resistance of 250 ohms requires (on the face of it) an emitter resistor of 5 ohms but, this is where things start to get tricky. For any BJT, \$r_E\$ (the internal emitter resistance) needs to be lower than maybe 1 ohm to get anything like a reliable gain of 50 (with an external emitter resistor of 4 ohms).
To find \$r_E\$ at ambient temperature, it is 26 mV divided by Ic so, to get that down to 1 ohm requires an Ic of 26 mA. Then, if you calculate how to get 26 mA from a 5 volt rail you need a resistance of 192 ohms and, of course, that is less than the collector load resistor so it's a no-go situation.
However, you could try and simulate it with virtually no emitter resistor to see how things shape-up but, for sure, the output distortion will be sub-optimal to say the least and the amplifier gain will be very temperature dependent.