Yes, you can compost beetroot – but not pickled ones.
Unpeeled beets can be quite hardy (since they’re used to living in soil) so chop them up to speed up the composting process, especially if the roots are fairly large/woody.
The leaves from beetroot plants can be composted too – but if they’re fresh, they can be eaten. Baby leaves can be eaten as a salad leaf while bigger ones, with their prominent stem, can be boiled, steamed or stir fried.
Avoid composting pickled beetroot – it’ll rot down pretty quickly but the highly acidic vinegar might cause problems for your compost heap. (It is, of course, used to preserve veggies by protecting them from the molds and bacteria that cause decomposition — in a compost heap, those guys are our friends!)