The earliest viable long-distance electrical communications system was the telegraph and its introduction spawned a whole range of new studies, techniques and products intended to maximise its benefits and its efficiency. New industries, producing long lengths of ductile, corrosion resistant wire, termite resistant wooden poles and ceramic insulators were born, and their successors are amongst the largest corporations of the 21st century.
It is a plausible piece of folklore that the choice of wire thickness and position of the insulators on the ubiquitous 'telegraph poles' was such that the characteristic impedance of the resulting transmission lines was 600 ohms. It is certainly true that the characteristic impedance of a wire-pair transmission line is a function of wire thickness, distance between the conductors and the permittivity of the insulation between the pair of wires and if you insert the typical dimensions used in early telegraph poles into that function, the figure of 600 ohms is a likely outcome.
Whether 'folklore' or not, 600 ohms was widely adopted as the 'standard' for telecommunications systems and later broadcast studio installations. Thus sending and receiving equipment and the cables linking them all had to work at 600 ohms, providing optimisation of signal-transfer and consistency in the operation of hybrid transformers used to provide the separation between the relative sending and receiving levels experienced by telephone users. In practice, more modern multi-paired cables had characteristic impedance closer to 140 ohms, resulting in the widespread use (in transmission line applications) of transformers (typically 2:1 ratio) to convert this to 600 ohms.
600 ohm sending and receiving impedance had been adopted by the broadcasters from the beginning of the industry in the 1920s and continued as the industry and its infrastructure grew. During the 1960s, in a period which saw further growth in radio and television output and facilities, it became apparent that the sheer volume of cable installed in the main distribution frames in the large broadcasting centres had effectively generated a large value of self capacitance which, in conjunction with 600 ohm impedance, began to manifest itself in noticeably poor frequency response. As a quick fix, studio sending impedances were dropped to typically 50 ohms and eventually the practice of sending from 'low' impedance studios into high or 'bridging' loads was universally adopted for subsequent analogue use without, it should be noted, any change in the cable infrastructure. The widespread use of 600 ohm systems persisted for much longer in telecommunications.