Without context, it is impossible to tell accurately. However, I would say it is likely that it means a sorrel(-coloured) animal. The first phrase is correct while the second phrase is not. Hope this helps :)

"Sorrel" can be a color, and often used of brownish-colored animals. "Beast" here is probably either a horse--an ordinary everyday workhorse--or an ox. "Crime and Punishment," right? (Checking) Yes, it's a horse. A Google image search on "sorrel horse" will show you dozens of pictures, and you can see what color "sorrel" is. We know it's a horse for two reasons. 1) He says "...a thin little sorrel beast, one of those peasants’ nags..." and a "nag" is a weak, old, tired-out horse. 2) He's drawing a contrast between big, heavy carts drawn by big, strong horses, and this one, which is drawn by a small, thin "beast." "In the shafts of such cart" is simply incorrect. You need the indefinite article. First he talks about big, heavy carts pulled by big, strong horses. "In the shafts of such a cart" means "a big cart like those other ones." It should be drawn by a big, strong horse. Instead, it is being drawn by a small, thin, "nag"--a weak, tired-out, small, old horse.