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Of course you can! You can do pretty much anything online. Well, unless you live in Vietnam. But if you lived there you probably wouldn't be doing GCSEs.
Well, actually I fibbed a little bit. You can certainly study for retakes at home, either on your own volition (if you're feeling particularly confident) or through any number of distance learning sites. You can downloaded syllabuses from the exam board's sites (along with old exams to practice with), but for the exams themselves, afraid not.
Which makes sense, doesn't it? There's no real way to make sure you don't cheat when you're working at home. So more often than not you'll have to arrange to take the exams themselves at a local school or adult learning service.
Of course that doesn't come cheap. If you try to teach yourself the syllabus, that's free, but the taking the exams again will cost you. Depending on the exam boards this could set you back between £10 and nearly £50.