Welcome EarlyMilly1.
Sputnik spoken here!
Sputnik transmitted a beep beep beeping sound at 20.005-40.001 MHz. The beeping can be easily modeled with any microprocessor or even CMOS circuit. The sticky part comes into transmitting. There are a few frequencies in that range that fit under the amateur satellite/amateur bands. You will need an amateur radio license to broadcast in this range.
21.0-21.45 Mhz
24.89-24.99 MHz
28.0-29.7 MHz
You could use the Arduino to control a DDS chip to generate 28.5Mhz. Mix in the beeping filter and transmit. I've grossly oversimplified this process but showing that it is doable.
Get a HAM license first.