Finally got a chance to view some of the LOC newspaper archives articles about the "Eureka Baseball Club". Here's the 1883-84 line-up:

J. Sullivan, captain 1883
T. Meagher 1883
W. Renfro 1883-84, pitcher
W. McLaughlin 1883
E. Furness 1883-84, 2nd base
R. Barry 1883-84, 3rd base
T. Costello 1883-84, short stop
J. Leonard 1883-84, 1st base
G. Hilbert 1883-84, rightfield
Crone 1884, centerfield
Mack 1884, catcher
Perry 1884, leftfield

This may have nothing to do with your image, but I found it interesting. It's possible the team in Sacramento was known as the Eurekas but I can find no record of it.

Anyways, managed to piece this together from a Sacramento Daily Record-Union article August 8, 1883. "The Eureka Baseball Club held a meeting last evening, and organized with the following members....". They played before as Eureka Baseball Club in May so I don't know why they "formed an organization" in August. And another July 21, 1884 SDRU article about a 14 to 0 victory over the "Peruvian Bitters". No stats, or particulars about players, but think I can find more info on players through 1880 census, or by searching names in archives.