Born in Hobart on 4 September 1895, Eric Claude Beedham was the youngest child (and second son) of Frederick Wescott Beedham and the former Isabel Sara Kipling.  He was educated at the Friends' High School (a denominational Quaker school, founded in Hobart in 1887) and later attended the short-lived Queen's College (a private school, which operated from 1893 to 1912).  Eric commenced his professional career as an articled pupil of his elder brother, Lancelot Kipling Beedham (1889-1944), who had established an architectural practice in Hobart in 1912.  Completing his articles in 1916, the younger Beedham joined the Hobart City Council as an Assistant Engineer & Architect.  During this period, he undertook at least one private commission under his own name - the design of a church hall in Goulburn Street, Hobart (1918), for which the then 23-year-old Beedham was credited as "honorary architect". Later that year, Beedham moved to Sydney and joined the office of John & H G Kirkpatrick, which, at that time, was the preferred architectural firm of the Commonwealth Government.  Beedham rose to the position of Managing Assistant for Australia, which required him to oversee the firm's projects undertaken outside New South Wales.  In that capacity, Eric Beedham supervised the construction of the Banknote Printing Office in Fitzroy (1923) and the new Commonwealth Bank buildings in Collins Street, Melbourne (1920-24), Moorabool Street, Geelong (1925) and Queen Street, Brisbane (1927-28). Beedham was also involved in the development of the Commonwealth's ambitious War Service Homes programme to provide architect-designed dwellings for returned servicemen.  
  
Concurrently, Eric Beedham had maintained a private architectural practice in both Sydney and Melbourne.  He was was still residing in Sydney in 1923 (when he married Nellie Godfrey) but had settled permanently to Melbourne by 1927.  There, his private practice initially concentrated on residential commissions.  One large house in Toorak, designed in 1925 for prominent businesman Louis C Moore, attracted much press attention both locally and interstate.  In early 1926, the Hobart Mercury reported that "news has been received in Hobart of the success on the mainland of Mr Eric C Beedham, FLIA, a young architect and native of Hobart.  He has received very considerable commendation for his design of the residence of Mr Louis C Moore, Byroyna, of Toorak.  The Australian Home Beautiful of December 12, 1925, has many illustrations of this residence, and speaks in glowing terms of Mr Beedham, one of Melbourne's youngest architects, who has set his mark on Toorak and is likely to be heard a good deal of in the future".  This would indeed be the case.

Eric Beedham's professional practice was still thriving by 1930, although he had already begun to move away from residential projects towards larger-scale commercial and industrial ones.  In 1934, he entered into partnership with architect John W Wright (1892-1962), who had maintained his own practice in Melbourne from 1921.  In partnership for about four years, Beedham & Wright undertook a string of commissions of various types, although the bulk of their output was factories - most of which, moreover, were located in the booming industrial heartland of South Melbourne.  These were invariably designed in the then-fashionable Streamlined Moderne style, with multi-paned strip windows alternating with continuous spandrels to provide a dominant horizontal emphasis.  By mid-1937, Beedham was again practising on his own, although he continued to specialise in factory design.  As was typical for many, Beedham's architectural practice diminished during the later years of the Second World War, but, unlike some, it re-emerged afterwards, when he resumed designing factories in Melbourne's inner industrial suburbs.  These, however, proved to be his last projects.  On the evening of 9 May 1947, Beedham collapsed at St Kilda Railway Station, and died as he was being taken to the Alfred Hospital.  He was 52 years old.  Beedham's elder brother, Lancelot, who had followed him to Melbourne in 1922 and gained a position in the Public Works Department, predeceased him by four years.