I want to be a ship captain (not an admiral or space general or commander), and explore and fight and just wipe out the enemies that stand up to me. And I want to do that with one ship, make a profit off of it. I don't want to manage a fleet and I don't want to worry about pre-battle deployment and all that 'noise'. It's fun don't get me wrong, but I wanna feel like a BADASS (and I don't trust the teammate commander AI with my life let alone a bunch of virtual spaceships). But with CR, I don't know what to do... I don't even look to refitting my ships. I just look at the cargo to see if I have enough space to buy more things to trade.
Back to the OP...
This is doable at the endgame, after accumulating enough skills and resources to store, but it is hard! With the right flagship, it can be sustainable, but it is nowhere near as rewarding as other income generators because you will be forced to leave most loot behind.
I tried out some ships by themselves, with no other ships in the fleet. (Exceptions: Odyssey and Heron got fighters as long as fleet had less than 40 DP.) I roamed the sector from bounty to bounty to see how long I can last. In any case, you have so little capacity that you can only loot supplies and fuel. Low capacity also means you cannot waste time chasing things other than your destination before supplies run out.
Of frigates, only Hyperion has a chance to solo nearly any fleet, but it cannot carry enough supplies to fix damage and recover CR. Hyperion will need to go to stations to buy supplies and repair damage and CR. Hyperion is great as part of a fleet, but terrible by itself. You will lose money with a solo Hyperion.
None of the destroyers (except maybe perfectly piloted Sunder) are powerful enough to destroy every possible bounty fleet. Medusa cannot handle high-tech capitals with Combat 10. Sunder might, but it is sluggish and has poor point defense, which means fighters backed by carriers can be a major problem. Enforcer and Hammerhead have limited ammo. However, destroyers can handle some bounty fleets, and they can loot more than they consume if they do not take too much damage. Just be prepared to avoid fleets your destroyer cannot defeat.
Cruisers might be the best option, and of all the cruisers, Aurora with Missile Specialization 10 and lots of Reapers is probably to the easiest cruiser to solo the game with. About the only ship it might not be able to kill is a max Combat Paragon (which I did not fight yet with solo Aurora). As long as the cruiser does not take too much damage, it can loot a bit more than it can use. However, one bad fight where your ship takes much damage and you can kiss your accumulated profits from previous battles goodbye.
Onslaught and Paragon can solo fleets, but they eat too many supplies to be sustainable. At best, a battleship might be able to barely come out ahead, but if it takes damage, you will need to go to a station to buy supplies and repair. They also cannot afford to pick on smaller fleets because they may not generate enough supplies to loot.
Carriers with fighters are not as efficient as dedicated warships. Heron with few fighters is incapable of soloing every possible fleet. Odyssey can solo fleets without a max Combat battleship, but not as efficiently as others.