Although intended to query the spacial existence of em waves I hope to draw closer to the mental processes involved and the reasons for the processes.

There are two major theories to consider: a) Transparency: in that light passes through an object, read radio waves pass through air and brick and b) if light does at all travel or if there is a coincidence of apparent transmission and reception as in the quantum state that if one changes all must change. This later quantum argument is extreme and goes to the heart of all arguments.

Transparency Apart from the common concepts of light passing through glass and water, - no doubt crystals were thought of as glass like and modern plastics are another matter - there is the reception of radio waves inside an apparently opaque structure.

Light is a variety of electromagnetic waves that the eye responds to by sending signal to an interpreter in the brain; these frequencies are high compared to radio signals but essentially are no different that the fact that high and low notes in music are just different frequencies of air pressure changes on the ear drum.

Radio waves, in the case of the BBC world service transmitted from Rugby, used a frequency of 15,000 cycles per second, the modern term is 15KHz (15 kilo hertz)
Television is in the range of 100's of megahertz
Mobile Phone around the 0.9 to 1.9 gigahertz
Bluetooth and wifi 2 to 6 gigahertz

So we have gone from generating electromagnetic waves of 15K that if it were sound would be audible to many people, especially the young to frequencies a million times that.
It may be easier to compare by the length of a wave. EM waves travel at a constant speed in a vacuum and that is the usual figure used. So a frequency that oscillates twice as fast will have a wavelength half the size.
Rugby (15KHz) 20Km
BBC Caradon Hill Digital (482.166 or 530.166)Mhz around 60cm

EE on 800Mhz (Rural) 37.5cm on 2.6GHz (London) 11.5cm
http://www.4g.co.uk/4g-frequencies-uk-need-know/

The Wi-Fi standard IEEE 802.11ad (60Ghz) 5mm
Active Denial System (ADS) which emits a beam of radiation with a wavelength of 3 mm
https://en.wikipedia.org/wiki/Extremely_high_frequency