Make a ASCII Hexagon Ring Tiling

Lua, 174 176 156 bytes

The code would print height-amount of lines, not height-amount of hexagons. Added *4, that fixed it, but added 2 extra bytes. Saved some bytes by changing the counter from if to modulo, and by putting two io.read()s into one.

Uses io.read() as input.

a,w,h,n,d={[[ /\__/\ \__/]],[[/_/  \_\/__\\]],[[\ \__/ /\__/]],[[_\/__\/_/  \\]]},io.read(,),"",0 for i=h*4,0,-1 do d=d+1 d=5%d end n=a[d]print(n:rep(w))end

Replicates the strings width-amount of times via string:rep(width), then iterates height-amount of times with a for-loop. Needed [[]] (literal strings) because the backslashes really screwed stuff up.

Old Version:

a,w,h,n,d={[[ /\__/\ \__/]],[[/_/  \_\/__\\]],[[\ \__/ /\__/]],[[_\/__\/_/  \\]]},io.read(),io.read(),"",0 for i=h*4,0,-1 do d=d+1 d=5%d end n=a[d]print(n:rep(w))end

Befunge, 137 bytes

I seem to have created some kind of scifi hand gun.

&>4*>~>$&>\>1-:4%3v
>00gg,\1-:v^_@#:\<>+00p\::6*\00g2/+2%+1+66+:
^%+66<:+1\_$$55+,^
_\/__\/_/  \
\ \__/ /\__/\
/_/  \_\/__\/
 /\__/\ \__/

Try it online!

Explanation

&>4*>                    Read the height and multiply by 4.
     ~>$                 Drop the 'x' separator. 
        &>\              Read the width and swap below the height

>                        Start of the outer loop.
 1-                      Decrement the height.              
   :4%3+00p              Calculate the pattern y offset: height%4+3
           \             Swap the width back to the top.

::6*\00g2/+2%+1+         Calculate the line length: w*6+(w+y/2)%2+1 
                66+:     Initialise the pattern x offset to 12, and duplicate it.

>                        Start of the inner loop.     
 00gg                    Get a pattern char using the x on the stack and the saved y.
     ,                   Output the char.
      \1-                Swap the line length to the top of the stack and decrement it.
         :v              Check if it's zero, and if so...
          _                 ...break out of the loop to the right.
       +1\               Otherwise swap the x offset back to the top and increment it.
 %+66<:                  Mod it with 12 to make sure it's in range.
^                        Then return to the beginning of the inner loop.

$$                       Drop the x offset and line length.
  55+,                   Output a newline.
     \<                  Swap the height back to the top of the stack.
 _@#:                    Check if it's zero, and exit if that's the case.
^                        Otherwise return to the start of the outer loop.