Yes of course, it can be done cheaply at the lower end. Our club is about 150 per year for membership. But if you want to compete...
Say you stay purely on the coastal circuit. Each weekend regatta is probably £100 per child all considered. Not including accommodation etc if required.
However if you move more towards fine boats, and want to compete at a higher level:
Each British Rowing event has higher entry fees than coastal for example, then they tend to only be held in Nottingham or Dorney. So you have a couple of nights accommodation each time etc. We spend on average £300 per event per child for those events, of which there can be 8 or so through the season they want to attend.
If you get better still and need your own boat (sculls are designed around weight brackets, most small clubs like ours don't have multiple across all brackets) then like anything, there is a range. When we looked for DD because she is a lightweight there weren't many second hand around, so we bought her a new one...at the lower end of the price range. Hers was £8k, plus blades, bags etc, up to around £9k. From a good brand, but not the pinnacle. It is unlikely she will outgrow it weight wise, but may want to 'move up' as t some point. She was 15 when we got it and had been rowing for about 3 years by then.
Ds will be at the same point next year, he may or may not be as in to single sculling as she is, if not we will hold off, and perhaps he uses hers periodically for training with a few adjustments. Yes, boats can be shared, but one of the main draws of having your own when you train as much as they do is that it is set up around your frame, he is 4 inches taller than her already at 14 and lighter still, so very different physically.
Add to that the speedcoaches at £500 each, the kit, the training camps, the erg for home training, the fuel, gym membership, trials etc.
So like most things, it can be done cheaply for sure, but if they get the bug and are good, they can quite quickly start costing £££.