You shouldn't need to turn anything off.
The heat source for the hot water will pump the same water that goes round the radiators through a coil/heat exchanger in the hot water tank to heat the water you actually draw off indirectly. This pumped water also goes trhough the rail if either HW or CH or both are on. Your example is how it should be done, many (mine included) are so far from the heat source it's only convenient to put them on the radiator circuit.
Close both valves (you will need to turn E by fitting a valve head (eg the one the other side) or using a spanner.
DO NOT MOVE A, G, F, B, D. A is a plug in case you want to fit pipes from the side.
There should be a bleed screw at the top of the rail. Open this - if a lot off water comes out, you haven't shut the valve(s). Otherwise leave open.
Undo one C with a bucket underneath (on top of a towel to protect the floor). When the water starts to dribble, carry on opening until a useful flow is obtained - you are now draining the towel rail, not the system.
When the water stops coming out, undo it completely and then the other C.
Remove the support fixings and the rail should come away from the wall.
Don't let children play with the valves!
Paint
Refit, doing up the C's hand tight plus 1/4 - 1/2 a turn (DO NOT OVERTIGHTEN).
Open one valve - water will rush in and air out the bleed valve, which you have left open.
Shut the bleed valve. There is now water in the bottom of the rail and you are testing for leaks. Assuming there aren't any, bleed all the air out. If there are try a little extra tighten of the appropriate C. If it's a disastrous leak (it shouldn't be) shut the valves, leave a bucket underneath and get someone else to have a look!