During the late 1960s and early 1970s he worked as a labourer and musician, with frequent forays to Chester, Crewe and other locations to perform at folk clubs and underground venues of the period.
He came to prominence in the early 1980s, with the publication of two volumes of poetry – mainly written in free verse – Noethni in 1983 and Jazz yn y nos three years later.
In the same period, he began to apply his poetry to music, performing as a singer and guitarist, initially with his Triawd (Trio), which evolved into his current backing group, Rhai Pobl (Some People).
His band has always included some of the most talented musicians on the Welsh language music scene, and currently includes Eaves' long-time collaborator, multi-instrumentalist Elwyn Williams, drummer Gwyn 'Maffia' Jones, double bassist Pete Walton and eminent folk musician Stephen Rees, as well as Eaves' daughter, Manon Steffan Ros.
The first is an honorary doctorate of the Open University (D.U., Open), awarded in 2014 in recognition of his ‘exceptional contribution to Welsh language and culture.’ The second is a PhD awarded by Cardiff University in 2015 for a thesis in Welsh, arguing for the key role of 'critical linguistic awareness' in the inclusive model of language planning currently pursued according to public policy in Wales.