Referring to this How can you fly your own military jet in terms of altitude and acrobatics? how can you request a block of airspace to freely maneuver your private military jet (barrel rolls, vertical climb, pulling Gs, etc..). If that kind of block of airspace is indeed possible to be permitted, what are your limits flying your jet in it?
As @jamessqf points out, once the aircraft is no longer owned by the government1 there wouldn't be any regulatory difference between you requesting air work and, say, a Gulfstream acceptance or post-maintenance flight that needs to fly around at odd altitudes and attitudes to check out various systems. That said...
Below FL180 you can fly VFR as you like. Of course you still have see-and-avoid responsibility, which might be helped by any onboard targeting radar on your military jet but then again might not be.
If you want to go higher (and have a smaller chance of turning someone's Cessna into a hood ornament) you need an IFR flight plan. Ideally you'd ask someone for permission to enter a Military Operations Area (example) and/or Air Traffic Control Assigned Airspace (which is the same concept as a MOA but exists in Class A airspace). FAA Order 7400.10 will tell you who the controlling agency and using agency are for each MOA; if you wanted you could try cold-calling Albuquerque ARTCC and asking if they'd take the Mt. Dora MOA hot for you, but the Air Force's 27th Special Operations Wing is the using agency and the line controller probably won't help you unless you run it by the military beforehand.
Assuming you can't get clearance into special-use airspace your only other option is to request a quadrant clearance in conjunction with a block altitude assignment.
A quadrant clearance (7110.65 4–1–1i) allows you to fly between two specified radials of a navaid and within two defined radii (DME arcs). Depending on the performance characteristics of your jet you might need a significant amount of distance, e.g. between the 10-mile radius and 40-mile radius, in order to get a large enough slice of airspace to work in.
A block altitude assignment (7110.65 4–5–7g) is more commonplace and allows you to operate anywhere between the two indicated altitudes. Again you'd probably want a good eight-to-ten thousand feet or more.
For both the vertical and lateral considerations you'd probably have the most luck over the Western states (ZLC, western ZMP, ZAB, etc). You could also try going offshore as long as you checked to make sure the warning areas weren't hot.
1Or operated on behalf of the government; I've seen a couple N-numbered British/European fighters operate using tactical callsigns as aggressor aircraft for military training exercises.
There is nothing specific to operating a military jet or any other type of aircraft.
If you are in class E or G airspace, you can fly VFR and do whatever maneuvers you want without even talking to ATC, though you are still required to see and avoid other traffic. This is generally how student pilots practice their maneuvers, though much lower and slower than you’d be doing in a jet.
If you want to fly in higher airspace classes and get positive separation from other aircraft, you will need to fly IFR, which means requesting and complying with a clearance of some sort.
A sector (as in randomhead’s answer) is one option. Another is a radius: within X miles of a fix or some radial/distance. Both would be combined with a block altitude.
Actually getting such a clearance from ATC without prior agreement would likely be difficult, especially given the size of the area you’d need for a military jet. Better would be to call (or visit, if possible) and work with them to find an area or two that would meet your needs without adversely affecting other traffic flows. The controllers in that area would be briefed on what was agreed and likely given a name you would use to request it in shorthand rather than have to explain your request every time, and there would likely be other shorthand (such as “knock it off”) if they need you to stop maneuvering for some reason, like military, medevac or non-participating aircraft that come too close to you.
As far as where you’d likely find such an area, consult nearby IFR Low/High charts and look for large white areas or SUA with no airways going through them. You would likely only get to use SUA when it is cold (many run on fixed schedules, so it’s not that bad), but they are convenient areas that already have names and little civilian traffic.
If you discuss your plan with an ARTCC's Airspace office, you may be able to reserve something ahead of time. It would save the controller a bit of work.
If you, however, showed up on the frequency and wanted a block of airspace as part of an IFR clearance, I could accommodate you with a quadrant clearance around a NAVAID ("Cleared to fly the northeast quadrant of the BFF V-O-R from the 010 radial to the 085 radial between one-five and six-zero miles, maintain FL280 through FL310..."), or a boundary type clearance ("Cleared to fly between GRN NDB, VTN NDB, and TDD VOR, maintain 170 through FL260...") and I don't need to know all of the specifics as to why you want the airspace. You'd be more likely to get your wish in a low altitude environment (below FL240, in most areas), though, and in an area that doesn't have a lot of traffic.
It would be very difficult to get this in IFR conditions in the busier ARTCCs.
As mentioned in the other answers, if you do the maneuvers as a VFR aircraft, all you really have to do is follow VFR rules, and you don't have to talk to ATC.
The simple answer is that you don't get your own block of airspace to do aerobatics in your private turbojet.
You could fly into a military operations area, but even if authorized to use that airspace (and some private owners are) for that purpose, anyone flying VFR may transit that airspace without talking to a soul...which means you don't own it. You just use it...which is the same as being anywhere else.
Military aircraft maneuvering in mock engagements or aerobatics are operating under VFR. Civilian aircraft maneuvering are doing the same thing. They are responsible for their own traffic separation. Participating traffic may be alerted or the aircraft doing aerobatics may be alerted, but VFR traffic separation is not guaranteed. It's the responsibility of the pilot performing aerobatics to clear the airspace, and to ensure he or she doesn't create a hazard or a collision.