Replacing a stuck stopcock - what could possibly go wrong?
You'll need some way to shut off the water when replacing the valve - that may require calling your water supplier, depending on local rules (if there's an outside shutoff that they own, they may not want you operating it - varies.)
In the case that there is no outside shutoff, there are fancy units that will freeze a short section of pipe to plug it, but you'd probably need to call a plumber who owns one - not a practical DIY tool for realistic budgets.
The valve you have may only need service (cleaning and lubrication), but if you are going to the bother of getting the water shut off to service it, I would replace it with a 1/4 turn ball valve, as I have much less trouble with those for shutoff service (they tolerate sitting idle for years without becoming impossible to operate, since they have a very simple mechanism with few parts to corrode and get stuck.) If you have the sort of valve you show, operating it on a regular schedule (like once a month) is the best bet for it being able to operate when it's needed.