He began his medical studies in Saint-Lô and subsequently moved to Paris to specialize in pharmacology.
However, Eudes was drawn to a more radical ideology than the humanism of Hugo or the Mutualist doctrine of the followers of Pierre-Joseph Proudhon, who dominated the French section of the International.
His associates included radicals like Ernest Granger, Gustave Tridon and Anne and Victor Jaclard.
In August 1870, Eudes was one of the ringleaders of an unsuccessful Blanquit insurrection at La Villette, a district of Paris.
Eudes was vehemently opposed to the peace negotiations undertaken by the new republican government of Adolphe Thiers.
On 24 March the Central Committee of the National Guard appointed Eudes Commissioner of War, along with Émile-Victor Duval and Paul Antoine Brunel.
In May, Eudes voted with the majority to establish a Committee of Public Safety, modelled on that of the first French Revolution, and became one of its members.
Eudes participated actively in the fighting during the Bloody Week (Semaine sanglante), 22–28 May, that marked the Paris Commune's last stand.
On 5 August 1888, Eudes was holding a particularly vehement speech at a meeting at the Salle Favié, when he suffered a cerebral haemorrhage due to an aneurism and died.