Are there hypothetical processes which allow the existence of a fission powered star? I know this is impractical given the rarity of heavy fissile elements in the universe (contrary to the abundance of fusion friendly elements like hydrogen), but is there any process via which a 'stable' star-like body could form, which shines because of fission?
(I can't picture how nuclear fission and gravity could 'self sustain' like fusion and gravity in typical stars, but the one thing I can say is that the stars lifetime would have to be lesser than the time it takes for a majority of its constituents to decay, but definitely longer than a typical chain reaction.)
 A: In principle yes, though it would be a highly contrived situation and not one likely to arise naturally.
A star works because hydrogen to helium fusion is energetically favourable. But the process has a huge activation energy so you need an environment as hot and dense as a star's core to provide that activation energy.
Likewise, for any sufficiently heavy nucleus fission is energetically favourable, but again there is (usually) a large activation energy. In fission reactors or bombs that energy is supplied by neutrons and exploits a specific pathway available to uranium-235. However any sufficiently heavy nucleus will fission if you put it in an environment where lots of energy is available. For example lead nuclei could in principle be fissioned and this would release energy. The trouble is that the energies required are of order 10MeV, and would require a temperature of around 10$^{11}$ degrees. While this is easily achieved in an accelerator it doesn't seem likely you'd manage this by assembling a large ball of lead.
A: As I see it, and, correct me if I'm wrong, but there is a way to do it.
The problem with using pure Uranium or any other readily fissionable element is that, as one element decays, that releases 2 neutrons which can speed up the decay of nearby elements.   If you had a ball of U235 or U238, say, the size of a planet or even a small moon, you'd have a cascade reaction and something resembling a bomb, like a small nova, not a star.   
There was a question very similar to this, I remember, but I can't find it.   I answered it, perhaps badly, but having answered it, I've given it some thought since then.    
The way to do it would be a mix of elements.   I think Uranium and Tungsten would be best - with more Tungsten than Uranium, I'd guess at least 10 parts to 1, maybe more and preferably Tungsten 182, cause 182, 183, 184 are all stable, so it could absorb neutrons without becoming radioactive itself.
The density of Uranium and Tungsten are quite similar, with Uranium slightly more dense.   We know from the temperature of the earth that radioactive decay heats the interior of the planet, so if we have a planet or star sized object that's maybe 3% or 5% uranium, that's going to generate a lot of heat, and likely create circulation and over time, gradually, the Uranium would fall towards the core and as it grows more concentrated, which would happen slowly, you would see an increased rate of fission.
I see no reason why a planet sized ball of Tungsten 182 mixed with Uranium couldn't be star like, though it wouldn't burn as long, but you might be able to create a surface temperature rivaling a star for a million or a few million years or so.
That would never happen in reality, it's just a theoretical possibility.
