What unit/technique is the best way to capture cities in the final age of Civ 5? I have been pumping out Death Machines but they don't really do very well versus cities and bombs.
You'll get the ultimate end-game unit a few technologies earlier than the Giant Death Robots you're using now: Stealth Bombers. They have a range of 20, and a strength sufficient to damage any unit in the game. Better yet, each of their attacks is a melee attack, so they gain XP quickly. They'll take some damage every time they attack, but you never need to worry about them being destroyed: they're sitting pretty at least 10 hexes away from the action, so when they're low on health, have them rest for a few turns. Before too long, your stealth bombers will have both Logistics and Air Repair, attacking twice a turn and healing 20 HP at the end of the turn in their cozy cities far from the front lines.
Once you have a sufficient force of Stealth Bombers, the hardest part is marching your melee units fast enough to capture the enemy cities that your bombers are blowing to bits. Any enemy units will be obliterated by your flock of bombers as you walk your melee troops into city after city. Use your Giant Death Robots to take the cities; if you're having problems with enemy air units, try using Mobile SAMs, Paratroopers, or fast units like Mechanized Infantry or Modern Armor instead. (Mobile SAMs have defensive bonuses against air units, and Paratroopers can be air-dropped next to the city, so you only need to take one turn of bombardment before capturing the city.)
If you're being nuked, then try to pillage your enemy's uranium mines so they can't build any more nukes. Strip away the defending units with your bombers, then march in your troops to destroy the mines. Or simply capture the cities that have the mines, using your overwhelming might of your Stealth Bombers. Did I mention that Stealth Bombers are powerful?
(Side note: Stealth Bombers take about a dozen seconds to attack, which gets time-consuming once you have several of them. To speed up the process of attacking, switch to the Strategic Map first.)
Artillery units (Rocket Artillery included) or (Stealth) Bombers can soften up a city without having to get particularly close, and then fast moving front-line troops like Mechanized Infantry can move in for the kill.
Against enemy air units, you may want to have Fighters on Air Sweep, or anti-air traveling along side your ground units.
When it comes to nuclear weapons, the best defense is to take them out of the picture by pillaging Uranium mines. There's not much you can do against enemy nuclear weapons otherwise.