Can civilians fly military aircraft?
I'm not sure if you're asking about flying such an aircraft (e.g. renting one) or owning one. But either way the answer is yes.

I've flown a Hawker Hunter at a (now-defunct) location in Cape Town on a mixed low-level/acrobatic flight along the coast. There were no restrictions on who could fly in the aircraft apart from the obvious ones (no kids; must have reasonable height/weight and general fitness) but they didn't let non-pilots take the controls. We did some aerobatics (up to 5G, from memory) and I had the controls in cruise flight for about 10 minutes. There was absolutely nothing that a civilian couldn't learn: it's 'just' another aircraft.

As for owning one, it's not uncommon at all. Privately owned ex-military aircraft are often referred to as "warbirds" and in the US there's even an EAA organization for owners. The aircraft are usually considered experimental (or the local equivalent) for the purposes of regulation and airworthiness.

The barriers to owning these aircraft are (as always) money and training. Buying and maintaining them is very expensive because there are so few parts or mechanics available, and insuring them is (almost) impossible so most owners have to self-insure. The operating costs are also usually high, especially if the aircraft systems need 'exotic' fuels, gases, lubricants or other consumables. You may need to include explosives on the list, in the case of ejection seats.

Training is a similar issue because so few people can instruct you in how to fly a military aircraft, but obviously this isn't a total barrier and I assume that if you're involved in that part of the flying community then you'll find the contacts you need.

So in general, if you can buy a military aircraft legally in your country, if you have enough money for operations and maintenance, if you can find someone to train you, and if you can comply with your local regulations on non-type (experimental) aircraft then you can absolutely do it. But for most people that's a lot of 'ifs' :-)