In nethack, you can't save and reload if you die. I want to do this.
Is there a distribution of nethack that will let me?
You should be able to save in NetHack by pressing Shift-S, but your save game is usually deleted again when you load it. You can get round this by simply backing it up into another directory (it'll be in your main NetHack directory and will most likely have the helpful ".NetHack-saved-game" file extension).
Some people disapprove of this sort of thing apparently, but frankly life's too short to worry about that.
More info here.
As agent86 said, there is what's akin to a "cheat mode" mode built-in to NetHack largely for testing, but you can certainly use it to play. It's called wizard mode:
From the excellent NetHack wiki, read up on its features there.
In wizard mode you are prompted if you wish to avoid death when you would otherwise normally succumb. Other prompts are different as well, for example random teleportation becomes controlled teleportation and you can wish for things not normally possible (even when the wish is fairly gotten). Messages may also reveal more information than otherwise.
It certainly isn't the true game, but it is reasonably close depending on your intentions. Permadeath and difficulty are enormous components to the gameplay design, so playing without them isn't quite the same game.
I have written extensive code for a NetHack tool and have taken to playing a "normal game" in Wizard mode lately to reproduce events when it pertains to my programming. I've been finding it quite enjoyable, but that's because my joy stems from the coding. Still it's all about enjoying yourself.
Nethack is alongside many Roguelikes in requiring permadeath - there's no configuration option or other easy way to cheat this system. It's a design decision they made, and there's not a lot of love for "non-permadeath" in the roguelike scene.
Backing up your save file between saves can avoid this, which is considered cheating by many roguelike players. The term for this is save scumming. Some roguelikes (including versions of Nethack; it's been a while so I don't know if this is still the case) actually have anti-save scumming code which tries to prevent your ability to restore previous saves.
Since Nethack is open source, it's possible that you could make a modification to the code to remove the "delete save on death" feature, but I can't find any precompiled/supported variants of Nethack that do this by default. (As I said, it's viewed as cheating, and there are already cheating modes built into Nethack, so it's not a priority for most developers). If you were willing to recompile your own version of Nethack, I'd suggest looking at delete_savefiles() in files.c. Commenting out the contents of this function may prevent the deletion of your save on death, although I haven't verified this.
If you're willing to broaden your focus past Nethack, I'd suggest looking at Angband. There are many Angband variants, and some are easier than others. There's no protection against save scumming in the code, so backing up and restoring your save is a fairly trivial process. There are also difficulty adjustment options such as "stair scumming" which ensures that your randomly generated dungeon floors are more likely to contain artifacts and other useful items.
Beyond that, there are several graphical roguelikes that do not feature permadeath. Notable among them is Dungeons of Dredmor, which is available on Steam. Another that I've had good success recommending to roguelike fans is Final Fantasy Fables: Chocobo's Dungeon. Despite it's cute exterior, is as roguelike as roguelikes come.