The Spillway is a dam between Lake Carthage and a small river below. Usually in the summer it's dry with very little water running over it, so you can climb down to it and take in some really pretty views. It's a fun place to climb around and just to spend an afternoon outside, chatting.
This is a museum not about straw bales but made out of them. It's 'green' in the old-fashioned way and insulated with straw bales instead of 'modern' materials. The focus of the museum is on South Dakota history, especially that of Carthage. If you have relatives who grew up there, you're really into South Dakota history, or you're a fan of Christopher McCandless (the man from Into the Wild), you would love it. It's probably most exciting to visit during Straw Bale Days, the city's festival in August.
Lake Carthage is right near the town--just follow the signs once you're in Carthage. There's a nice swimming beach, and you can fish in the lake as well. There are also two places you can camp in Carthage--either near the lake or at Redstone Park on the edge of Main Street. See the website below.