Is it possible to make a 20 MHz radio with this
by creating a circuit consisting of a battery, that crystal oscillator and an antenna?
No. It's not an oscillator, it's just a crystal. You'll have to build an oscillator with it yourself, which only requires a few components, for instance like this one with a FET:
If you don't want to build the oscillator yourself you can also buy them for a few dollars. Then you don't have to worry about stability and such.
The oscillator won't give you enough power to drive a 50 Ω antenna, but an HF amplifier following the oscillator will do the job.
(*) The encased oscillator shown outputs a digital square wave, so you'll have to filter that, otherwise you'll transmit 20 MHz, 60 MHz, 100 MHz, etc.
If you really mean just an oscillator (the crystal is only part of one as Steven says), a battery and an antenna, then the answer is no, not a radio that will be of much use anyway.
All this would do is create a 20MHz carrier frequency which you will need to modulate somehow in order to transmit information. This could be as simple as a on/off switch (e.g. morse code) or some more complex form of modulation.
Knowing a bit more about what you actually want to transmit would help provide a better answer.