Draw Concentric ASCII Hexagons

Python - 251, 240, 239 228

l=input()+[0];m=max(l);A=abs;R=range
for j in R(2*m+1):print''.join([[' \\'[(A(j-i+m-1)/2.in l)*(2*m-i)/(j-m-.5)>1],'/'][(A(3*m-i-j)/2.in l)*(i-m-j+.5)/(j-.5-m)>0],'_'][(A(m-j)in l)*(A(2*m-i-.5)<A(m-j))]for i in R(4*m)).rstrip()

Alternative approach (251):

l=input()+[0]
l.sort()
m=max(l)
M=2*m
s=[[' ']*m*4for j in' '*(M+1)]
for i in l:
 I=2*i;s[m-i][M-i:M+i]=s[m+i][M-i:M+i]='_'*I
 for k in range(i):K=k+1;s[m-k][M-I+k]=s[m+K][M+I-K]='/';s[m-k][M+I-K]=s[m+K][M-I+k]='\\'
for t in s:print''.join(t).rstrip()

CJam, 148 116 109 bytes

This took a lot longer than I expected. Originally, I just wanted to iteratively build the upper left quadrant, as in the diamond challenges, and then get the rest from mirroring. But I didn't notice that the underscores don't obey mirror symmetry between the upper and lower half. So I had to redo most of that, to generate the right half iteratively and then only mirror once (to the left).

S]2[l~]:(f#:+2bW%{_,2/~:T;{IT):T1<'\'/?S?S++}%__,2/=,2/I'_S?*_S+a@+\I'/S?S++a+}fI{)T)2*2$,-*1$W%"\/"_W%er@N}/

Test it here.

A Fibonacci-esque example:

8 3 1 5 2
        ________________
       /                \
      /                  \
     /     __________     \
    /     /          \     \
   /     /   ______   \     \
  /     /   / ____ \   \     \
 /     /   / / __ \ \   \     \
/     /   / / /  \ \ \   \     \
\     \   \ \ \__/ / /   /     /
 \     \   \ \____/ /   /     /
  \     \   \______/   /     /
   \     \            /     /
    \     \__________/     /
     \                    /
      \                  /
       \________________/

Explanation:

As stated at the top, I start by building the right half iteratively. That is, initially I've got only a single space in the grid, and then for each possible ring, I either surround the existing grid in spaces or a new semi-hexagon.

Once that's done, I mirror each line to the left and pad it with leading spaces for correct alignment. Here is a breakdown of the code:

"Prepare the input and the grid:";
S]2[l~]:(f#:+2bW%
S]                "Push string with a space and wrap it in an array. This is the grid.";
  2               "Push a 2 for future use.";
   [l~]           "Read and evaluate the input, wrap it in an array.";
       :(         "Decrement each number by 1.";
         f#       "Map each number i to 2^i.";
           :+     "Sum them all up.";
             2b   "Get the base two representation.";
               W% "Reverse the array.":
"At this point, the stack has the proto-grid at the bottom, and an array of 1s and
 0s on top, which indicates for each hexagon if it's present or not.";

"Next is a for loop, which runs the block for each of those 0s and 1s, storing the
 actual value in I. This block adds the next semi-hexagon or spaces.";
{ ... }fI

"First, append two characters to all existing lines:";
_,2/~:T;{IT):T1<'\'/?S?S++}%
_                            "Duplicate the previous grid.";
 ,2/                         "Get its length, integer-divide by 2.";
    ~:T;                     "Get the bitwise complement and store it in T. Discard it.";
        {                 }% "Map this block onto each line of the grid.";
         I                   "Push the current hexagon flag for future use.";
          T):T               "Push T, increment, store the new value.";
              1<'\'/?        "If T is less than 1, push \, else push /.";
                     S?      "If the current flag is 0, replace by a space.";
                       S++   "Append a space and add it to the current line.";

"So for hexagons this appends '\ ' to the top half and '/ ' to the bottom half.
 For empty rings, it appends '  ' to all lines.";

"Now add a new line to the top and the bottom:"    
__,2/=,2/I'_S?*_S+a@+\I'/S?S++a+
__                               "Get two copies of the grid.";
  ,2/                            "Get its length, integer-divide by 2.";
     =                           "Get that line - this is always the middle line.";
      ,2/                        "Get ITS length, integer'divide by 2.";
         I'_S?*                  "Get a string of that many _ or spaces depending on the 
                                  current flag.";
               _S+               "Duplicate and a space.";
                  a@+            "Wrap in an array, pull up the grid, and prepend the line.";
                     \           "Swap with the other copy.";
                      I'/S?      "Choose between / and a space depending on the flag.";
                           S++   "Append a space, and add both characters to the line.";
                              a+ "Wrap in an array, and append line to the grid.";

"This is all. Rinse and repeat for all rings. The result will look something like this:

_____ 
     \ 
___   \ 
__ \   \ 
_ \ \   \ 
 \ \ \   \ 
_/ / /   / 
__/ /   / 
___/   / 
      / 
_____/ 

Note that there are still trailing spaces.";

"Finish up all lines. These will not be joined together any more, but simply left
 on the stack in pieces to printed out back-to-back at the end of the program.
 The following runs the given block for each line:";
{ ... } /

"This generates the necessary indentation, then mirrors the lines and puts them
 in the right order:"
)T)2*2$,-*\_W%"\/"_W%er\N
)                         "Slice off that trailing space, but leave it on the stack.";
 T                        "Remember T? That still has something like the the size of
                           the grid from the last iteration. In fact it's N-1, where
                           N is the largest visible hexagon. We can use that to figure
                           out how many spaces we need.";
  )2*                     "Increment and double.";
     2$                   "Copy the current line.";
       ,-                 "Subtract its length from 2*N.";
         *                "Repeat the space that often. This is our indentation.";
          \_              "Swap with the line and duplicate.";
            W%            "Reverse the line.";
              "\/"_W%er   "Replace slashes with backslashes and vice versa.";
                       \  "Swap with the original line.";
                        N "Push a line break.";

APL (222 bytes in UTF-8)

(and 133 chars)

Since this question specifically asks for amount of bytes in UTF8 representation, I had to actually ungolf it a bit so that it is longer but its UTF8 representation is shorter. (In particular, the commute operator's character is three bytes while () is only two, so that optimalisation doesn't work anymore, and it also makes assignment very expensive.)

{⎕←(~⌽∧\⌽⍵=' ')/⍵}¨↓⊃{⍵{⍺=' ':⍵⋄⍺}¨K↑(-.5×(K←⍴⍵)+⍴⍺)↑⍺}/{Z⍪⌽⊖Z←↑(⊂(⍵/' '),(2×⍵)/'-'),⍵{((-⍵)↑'/'),((2 4-.×⍵⍺)/' '),'\'}¨⌽⍳⍵}¨N[⍋N←,⎕]

Previous version, which is shorter in characters (124) but uses more bytes when represented in UTF-8 (230, which would put it in second place):

M←' '⋄{⎕←⍵/⍨~⌽∧\⌽⍵=M}¨↓⊃{⍵{⍺=M:⍵⋄⍺}¨K↑⍺↑⍨-.5×(K←⍴⍵)+⍴⍺}/{Z⍪⊖⌽Z←↑(⊂(⍵/M),'-'/⍨2×⍵),⍵{('/'↑⍨-⍵),'\',⍨M/⍨2 4-.×⍵⍺}¨⌽⍳⍵}¨N[⍋N←,⎕]

Test:

      {⎕←(~⌽∧\⌽⍵=' ')/⍵}¨↓⊃{⍵{⍺=' ':⍵⋄⍺}¨K↑(-.5×(K←⍴⍵)+⍴⍺)↑⍺}/{Z⍪⌽⊖Z←↑(⊂(⍵/' '),(2×⍵)/'-'),⍵{((-⍵)↑'/'),((2 4-.×⍵⍺)/' '),'\'}¨⌽⍳⍵}¨N[⍋N←,⎕]
⎕:
      3 1 5 2
     ----------
    /          \
   /   ------   \
  /   / ---- \   \
 /   / / -- \ \   \
/   / / /  \ \ \   \
\   \ \ \  / / /   /
 \   \ \ -- / /   /
  \   \ ---- /   /
   \   ------   /
    \          /
     ----------