Let's first make it clear: dogfighting in space is extremely unlikely with current technology due to immense fuel requirements: quick maneuvering -> extreme changes of impulse -> burn tons of fuel (comparable to mass of the craft) -> need to take more fuel onboard -> heavier craft -> need more fuel to change impulse... If there is no major breakthrough in high-impulse propulsion tech, there'll be no dogfights in space.
Also, orbital physics puts some serious constraints on possible maneuvers (if you played Kerbal Space Program, you know it), BUT it also makes quite awesome possibilities for hi-adrenaline low-orbit tactics without requiring tons of fuel (imagine Gravity with fighters instead of debris). If the planet is small and dense, its low orbit will look like a hi-speed circular death race track! (It'd be an interesting side-research to calculate which celestial bodies provide best conditions, or to choose physical parameters of a planet when designing a fictional one). As a bonus, it can provide "civilian background" to discourage from inaccurate shooting.
Also^2, here is some neat analysis on how physically realistic space battles may look like (you've probably read it, so just in case). And here is some fruitful discussion on dogfighting in space.
Finally, consider dogfighting inside a gas giant atmosphere. Ice giants' atmospheres contain more methane and ammonia than Jupiter-class giants, which can probably be used as fuel in an appropriate engine (that's a topic for a separate research).
Still sticking to dogfights? All right, let's keep it more or less hard-sciencey.
Current spacecrafts are extremely fragile. If a dangerous space debris is detected in the path of the ISS and it is too late to maneuver, the lids on Cupola are closed, solar panels are rotated in parallel, the crew gets into spacesuits and prepares for emergency evacuation in Soyuz. There is some layered shielding on the most vulnerable parts of the ISS, but it can only protect from small debris.
Armor is heavy and bulky. Even the battle-oriented spacecrafts of near future probably won't be strong enough to survive a direct hit of a missile, a volley of projectiles or piercing laser shots. Larger ships may afford some multilayer graphene shielding, but still, most of the time they'll bet on reflection, deflection and maneuvering ahead of time.
Spaceships are easily seen against space background, either visually, or in IR, or in radio. The closer we are, the easier it is to detect, and the harder it gets to stay stealth. A bit harder when against a planetary background.
The closer are the spaceships, the more weapon possibilities they have:
- Over large distances (light-minutes or more), only relativistic
weapons are feasible. It is a battle of AI-guided prediction
modeling, small randomized maneuvers and lots of patience. Assuming
we have a good low-dispersion hi-power laser/maser/particle-beam.
(Btw, consider a story with battles of AI predictors and an
exploited/faulty RNG which gives not-good-enough random maneuvering,
putting the ship in danger).
- Medium distances allow for mass drivers, which accelerate small projectiles to almost relativistic velocities. Deploy several
thousands of them into the predicted target position and around it.
And again, have some patience and a good prediction model/algorithm.
- Close encounters would allow using guided/smart missiles without waiting for months, but they are heavy to carry around and to
accelerate, can be countermeasured, and it is probably easier just to
deploy huge swarms of them from the mainship at a distance and leave
them do their job. Maybe even outfit them with their own beam
weaponry, turning it into a drone fight (which is also easier,
cheaper and safer for the crew, although less heroic), or shrink them
down into a micro/nanoswarm.
The closer are the spaceships, the fewer defense possibilities they have. Missiles, drones and other slow macroscopic objects can be taken down/redirected/blinded with beam weaponry ahead of time. Volleys of relativistic bullets are much harder to dodge, so we'll have to rely on random maneuvering and those puny armor sheets we have. Hence, mass-drivers can be used in both medium and close encounters (I just can't imagine an efficient countermeasure besides maneuvering, armoring and blinding/disabling enemy ship). As for beam weapons, besides maneuvering, we may try reflective (in appropriate wavelengths) surfaces, but another possibility exists: if there is only one source of enemy fire, we can shoot a deflector missile in their way, to change beam paths ahead of time (gravity lensing itself may be enough to make the beams miss). Or use dispersive cloud countermeasures.
Maneuvering, design and CM. Since maneuvering is the most reliable method of long-term survival in space battles of any distance (and since we have soften our fuel requirements), the smarter and more unpredictable we are, the better (AI can do the job, while keeping the crew safe in the mainship, but again, not too heroic). So, the ships should be light, be outfitted with enough reaction wheels to quickly turn around, may have several maneuver engines and several weapons looking in different directions, and be filled up with all kinds of countermeasures (jammers, optical/IR/MW/radio decoys, dispersive clouds, beam deviators, anti-missile beams, drones/nanoswarms, etc.) to get closer to each other. The ships should probably be of some fancy shape to make barrel rolls in 3D more efficient (no need to be aerodynamic). And those manned fighters should better be accompanied by numerous drones.
To sum it up:
There may be some situations where close encounters are possible (limited space inside some asteroid cave, for example) but there aren't many of them in outer space. Low-orbit is more feasible. Or maybe a scenario of protecting a space station/habitat from non-destructive invasion/boarding.
There are numerous technologies that would prevent ships from getting into a dogfight: algorithms of control and prediction, long-range beam weaponry, AI-controlled drones/swarms. This can be facilitated by prohibiting them in your scenario (religious fear of AI, "undignified" use of drones and long-range beams etc.).
And of course, some technologies are just not advanced yet to provide a dogfight: economical high-impulse propulsion (the most problematic one), efficient shielding, more rigid spaceships, cheap space launches / interplanetary travel, efficient weapons for space usage.
Also, remember: ships are small. Space is immensely huge. Consider sticking your spacefights to some locations of interest (as localized as possible; otherwise they'd just lazor down each other from afar).