With the correct rocket setup would it be possible to thrust around in a circle, like you would do donuts in a car, to generate centrifugal gravity?
Thrust won't generate gravity, but it will produce acceleration which may be indistinguishable from gravity to an occupant. Yes, it's possible to simulate gravity by having a spaceship constantly thrusting to travel in a circle, but it would be an awful waste of fuel.
Rotating a large object, or pair of objects connected by a tether, to simulate gravity is more efficient because, once started, it will continue to rotate with no further input of energy.
$\begingroup$ Also, while you can create artificial gravity by rotating the craft, you really need a large craft (torus shaped) or quite a long tether to get meaningful amounts of artifical gravity. The large craft scenario would mean ships immensely larger than anything we have yet constructed for space. The long tether is more practical but is tricky to get working without disrupting the orbit of the vehicle (and maintaining stability). It's also something you'd need to un-do before re-entry and any docking ships would need to match your rotation (if a space station). $\endgroup$ Nov 27, 2017 at 13:12
$\begingroup$ This all adds complexity and fuel requirements. All that said, most of the reason we send people to space (now) is to do microgravity experiments so any space-station implementation would need to have a rotating (gravity) section and a non-rotating (microgravity) section and a convenient way for astronauts to move between both areas without injuring themselves, damaging things, or just plain getting disoriented. As such, there hasn't really been a lot of effort put into actually making such a system, even though it's perfectly tractable on-paper. $\endgroup$ Nov 27, 2017 at 13:16
$\begingroup$ It is noteworthy that some of a "donut" movement would continue as well without further energy consumption. Part of the continuous thrust during a car donut exercise is needed to overcome friction (which is absent in space) and to prevent linear acceleration (which is undesired during the car trick but may actually be advantageous in a spaceship). The resulting movement after engine stop would be some tumbling in the original reference system but some rotation in the inertial system of the ship's center of mass. $\endgroup$ Nov 27, 2017 at 14:04
$\begingroup$ @J... large just means the gravity at your head is more similar to the gravity at your feet. If that doesn't matter you can just spin faster. $\endgroup$– fectinNov 28, 2017 at 3:43
1$\begingroup$ @fectin It's the "just spin faster" part that becomes problematic... $\endgroup$ Nov 28, 2017 at 4:28
For a space based object, there is absolutely no problem with the energy need - you can get that from solar panels, RTG (fission heat to electricity via thermoelectric effect). Each comes with a different set of tradeoffs, but they all work now in one spaceship or another (Mars rovers, satellites, deep space probes, ...).
What you can't have in space is reaction mass (as in Newton's action and reaction). Even the most mass-efficient thruster (electrically accelerated ion thrusters) use up their fuel - and in deep space you can't get any more (except for the - currently theoretical - Bussard Ramjet).
Also, any practical gravity (let's say half Earth gravity) is high in terms of fuel burn rate. I didn't find fuel burn values, but the trans-lunar injection was a 350 seconds burn for 3 km/s delta-v, or about one gravity. The fuel flow was (according to this page: https://www.nasa.gov/centers/marshall/pdf/499245main_J2_Engine_fs.pdf) about 530 lb/sec, or 250 kg per second. In monetary terms, that's more than half a million dolar per second - but the real issue might be one of availability not one of cost (rockets can only lift so much).
So, while the theoretical answer to your question is yes, the practical one right now is no.
Sure, though I can't think of a reason you'd want to do that. Burning the engines in any direction produces acceleration that feels like gravity. Given the tremendous cost of bringing more fuel, if you're in a spacecraft you are probably trying to get somewhere.
Spinning in a circle gets you nothing. There isn't a tether or similar to supply the centripetal force, so you'll be supplying it with the engines. And after you've spun around in a circle your net change in velocity is zero.
More likely, you'd just burn prograde or retrograde, in a straight-ish line, turning only to keep the ship aligned with your orbit. Besides using fuel to get you to where you are going rather than burning in circles, the occupants will be more comfortable as they won't need to deal with significant Coriolis forces.