However I do not really agree that storage heaters use power 'whether needed or not' - they are thermostatically controlled just like gas, so only come on when needed.
Only to a point. They turn on at night to heat a big pile of bricks - controlled by a thermostat which sets how hot to heat them, and hence ho much heat to store. During the day, they release the heat in a seemingly uncontrolled manner - I think the ones at work have a flap which may be just manually set or may have a thermostatic element to it.
The problem is that the flap arrangement isn't very good, and effectively you get heat out a) when you don't need it, and b) in a not very well controlled manner. So a portion of what you put in is wasted by either heating the room when you don't need it, or by over-heating the room (as I mentioned, we frequently have to open windows when the office gets too hot). In a domestic setting, you put a lot of heat in, it comes out during the day while you are out, and then in the evening they run out of heat.
Assuming of course that you aren't obsessive-compulsive enough to manually run them and keep adjusting the output control.
In theory, adding "proper" controls could largely mitigate this problem, but I don't know if anything available does that.
In theory a wet system with a water cylinder heated on off-peak could work better (as in how most people would be used to) - but to get the most out you'd need to oversize rads to work at low temperatures as the store depletes (from memory, dropping the supply temp from 70˚C to 50˚C roughly halves the output of a rad). You may need a sizeable store if you want a reasonable heat storage. You'd also want a thermostatic mixer valve to mix hot water from the store with cooler return water so as to a) avoid very hot rad temps when the store is "full", and b) keep a relatively uniform feed temp and so allow TRVs to work properly.
Say you worked between 90˚ when full and 50˚ 'empty' (ie 40˚ difference). It will take 4.2kW to heat 1l of water by 1˚C/s. So 4.2x40 = 168 kWs per litre of water. Thus you'd need 3600/168 = around 10l of water per kWHr of heat storage.
Work out your heat requirements in kW Hr / day, multiply by 10, and that's roughly what you'd need to do all your heating from stored heat. You could still have a second immersion heater to top up part of the store at day rates if you were to run out.
If that is all you use the store for, then any old cylinder will do. If you want hot water from it as well, then you are into thermal store/heat bank territory and can add another digit to the price
You'd also need a considerably bigger store.
The biggest factor that determines if this is practical is whether there is room for the store - closely followed by how hard/disruptive it would be to install all the plumbing.