World
Don't know what licence or experience you have, but yes it is possible to fly ex-mil jets even on a PPL.
If you are going to hire someone else's or get involved in a syndicate there will be minimum experience requirements for insurance purposes. In any case you'll need to do a conversion and then have the type entered specifically on your licence. Medical-wise, Class 2 would suffice.
In the UK at least ops tend to be limited to Day VFR only.
JP/L29 as mentioned probably your best bet to begin with. The difference in performance/fuel burn between the Mk3 and Mk4/5 JP is marked. Even a JP will go through fuel at a horrendous rate at low-level. Regardless of what you end-up flying, compared to light piston-engined aircraft you're practically on a fuel-emergency in a mil jet as soon as you take-off! Flying discipline needs to be high, you'll always need to fly with a diversion, even though VFR.
Hunters, Gnats are considerably more complex, and being swept-wing, much more likely to kill you if you have a bad day. Lots of systems to learn, pretty complicated emergency drills, Gnat in particular.
Finally, I wouldn't personally fly an ex-mil jet without a live ejection seat. Some in the UK have had the seats disarmed which severely limits your options should the donkey quit. Knowing when to make the decision to jump-out explosively is also handy, it has caught-out a few pilots from non-military backgrounds over the years...