Using lower frequency signals to transfer data
If the antenna size would be 1/20 of the wavelength would such a
  device be feasible or are there some considerations that make the idea
  impossible?


A monopole that is 1/20 \$\lambda\$ will have a radiation resistance of a few ohms and a capacitive reactance of about j1000 ohms: -



So your first fight is radiation resistance (equivalent load-resistance of what is transmitted as an EM wave) versus antenna loss-resistance and this latter may be of the same order as the radiation-resistance so immediately, you are wasting significant power in order to transmit power.

With a quarter \$\lambda\$ antenna the radiation-resistance is about 37 ohms so it's easier to deliver power to and has less percentage loss due to it being much higher than the loss-resistance.

If you use a resistor of (say) 40 ohms to produce a 50 ohm load on your transmitter (antenna is 5 ohms radiative and 5 ohm loss) you can see that 1 watt into 50 ohm (7 VRMS) becomes about 10 mW transmitted as an EM wave. Do the math!

This is a power loss of 40 dB so you have to think about using a transformer to effectively lower the output voltage from the transmitter in order to get a decent "match" to the 5 ohm radiation resistance.

At this point I haven't calculated radiation resistance - I've just used the graph above and made an estimate. So you should grab the formula from the internet and get a more likely number.

The next problem is tuning-out the capacitance of j1000 ohms. You would need a series inductor of equivalent reactance and this needs to be carefully chosen to make the circuit electrically resonate so that you can deliver power effectively.

So you'll need a transformer of about 3:1 turns ratio (50 ohm to 5 ohm) and an inductor of 1000 ohms to get a decent efficiency figure. It's doable at 10 MHz and should be OK but the antenna will be easily detuned by obstacles so you need to take care of moving objects around the antennas at both ends of the link.