The new ground of the St. Louis Union Club is said to be the smallest in the country. The second base will be very near in the center of the inclosure. Is is just seventy-one steps from right field fence to the home plate, or not to exceed 200 feet. Left field and center field may be fifty feet deeper. In no field, unless it be center does it seem possible for a batter to bat a ball for two bases with lively fielding.
-Cleveland Herald, February 6, 1884
While I'm not certain how accurate the Herald's description of the Union Grounds is, it's my understanding that right field was very shallow. The rest of the outfield, especially left-center, appears to have been rather deep. I'm sure that I'll find more information about the dimensions of the park as I get closer to the start of the 1884 season and the opening of the new park.
Below is the best image that I can find of the plaque that was dedicated at the site of the Union Grounds. I found it at the SABR website and it was taken by Joan Thomas, an expert on St. Louis ballparks. Joan is the author of St. Louis' Big League Ballparks and, I believe, was one of the driving forces behind getting these historical markers placed at the sites of St. Louis' old ballparks. If you take a good look at the image, you can see the short right field wall: