The Isle of Wight test facility is free to visit, it's worth a look if you're in the area. There isn't a massive amount left - just the concrete structure and a lot of rabbits grazing the cliffs below.

However, there's enough left to make you slightly sad that the space race didn't end up in colonisation (yet). There are also a bunch of other features within less than a mile (an old military battery, The Needles, great views over The Solent, the colored sands of Alum Bay). Definitely underrated.