How can I calculate the distance to the closest player?

Just in case anyone tried to look up how this problem could be accomplished, because this is the first result from google search. And I happened to work it out.

You can do this in 1.13+ now, with the /execute store command

The steps are as follows: store player coordinates > calculate the differences > square the differences > sum up > calculate the sqrt values. I'm using euclidean distance as the metric.

1. Store Player Coordinates:
You'd need to define objectives to store the values, I defined them as "X", "Y", and "Z"

execute as @a run execute store result score @s X run data get entity @s Pos[0]
execute as @a run execute store result score @s Y run data get entity @s Pos[1]
execute as @a run execute store result score @s Z run data get entity @s Pos[2]

2. Calculate the Differences:
I defined another set of objectives to store the results "X_difference", "Y_difference", "Z_difference", but you don't have to.

execute as @a run scoreboard players operation @s X_difference = @s X
execute at @a run scoreboard players operation @s X_difference -= @p[sort=nearest] X
execute as @a run scoreboard players operation @s Y_difference = @s Y
execute at @a run scoreboard players operation @s Y_difference -= @p[sort=nearest] Y
execute as @a run scoreboard players operation @s Z_difference = @s Z
execute at @a run scoreboard players operation @s Z_difference -= @p[sort=nearest] Z

3. Square the Differences:
   

execute as @a run scoreboard players operation @s X_difference *= @s X_difference
execute as @a run scoreboard players operation @s Y_difference *= @s Y_difference
execute as @a run scoreboard players operation @s Z_difference *= @s Z_difference

4. Sum Up:
   I defined another objective "distance" to store the sum of the squared values from 3.
   

execute as @a run scoreboard players operation @s distance = @s X_difference
execute as @a run scoreboard players operation @s distance += @s Y_difference
execute as @a run scoreboard players operation @s distance += @s Z_difference

5. Approximate the Square Root Values:
   We can use a simple algorithm to calculate the square root values. You can find more info on how it works here. We need to create three new objectives: sqrtI, sqrtX and realDistance. I highly recommend using a datapack to implement this, as we need to recursively get closer to the real square.

Once you have your distance value, run the following to reset the sqrt algorithm:

# ... every tick, after calculating distance, do:
scoreboard players set @a sqrtI 1
scoreboard players set @a sqrtX 0
scoreboard players set @a realDistance 0
# then we start the algorithm by running another function:
execute as @a if score @s distance > @s sqrtX run function mydatapack:sqrt

mydatapack:sqrt.mcfunction contains the following commands:

scoreboard players operation @s sqrtX += @s sqrtI
scoreboard players add @s realDistance 1
scoreboard players add @s sqrtI 2
execute as @s if score @s distance > @s sqrtX run function mydatapack:sqrt

The realDistance objective then contains the real distance that every single player is away from you in blocks.