Embedded Diamonds!

Charcoal, 28 24 23 bytes

↗²↖↙⁴⟲Ｐ²⁴⁶↘↙↙²⟲Ｐ²⁴⁶Ｃ⁸¦⁰

Try it online! Explanation:

↗²↖↙⁴   Draw the top left 4x4 corner
         /
        //
⟲Ｐ²⁴⁶   Rotate it three times to complete a diamond
         /\
        //\\
        \\//
         \/
↘↙↙²    Draw the top left line of the inner diamond
         /\
        //\\
        \\//
         \//
⟲Ｐ²⁴⁶   Rotate three times to complete five diamonds
Ｃ⁸¦⁰    Copy everything 8 characters to the right to complete eight diamonds

Edit: My previous answer used ‖Ｍ‖Ｍ↓ to create a whole diamond by reflecting the top corner, but due to a bug in Charcoal this left the cursor out of position. Fortunately I discovered another way of achieving the same effect which doesn't move the cursor, which meant that I saved a byte drawing the inner diamond. (I also looked into overlapping operations, but I wasn't able to improve on the rotate/copy approach.)

Edit: ↖²↗↘⁴‖Ｍ¬Ｍ¹¦³↘²‖Ｍ¬Ｃ⁸¦⁰ does the job in 21 bytes, but that relies on the new reflecting behaviour, so I don't know whether this was possible with the old code.

Charcoal, 62 50 48 45 42 40 bytes

Thanks to @Okx for saving two bytes!

↙²→↘²↙↖⁴→↗⁴‖ＭＦ²Ｃ⁸¦⁰Ｍ⁸↓→↗²↓↘²Ｍ⁴→↑↗²↓↘²‖Ｍ↓

Try it online!

Explanation:

↙²→↘²          // Draw the left half of the inner part of the first diamond.
↖⁴→↗⁴          // Draw the left half of the outer part.
‖Ｍ             // Mirror the half of the diamond to create the first full diamond.
Ｆ²Ｃ⁸¦⁰         // Copy the picture eight characters to the left twice.
               // The first copy gives us two diamonds, and the second copy of the
               // two diamonds overlaps so that the result is three diamonds.
Ｍ⁸↓→↗²↓↘²     // Move down to draw the upper inner half of the first middle diamond.
Ｍ⁴→↑↗²↓↘²     // Move right to do the same for the second middle diamond.
‖Ｍ↓           // Mirror the whole thing vertically.

PHP, 126 Bytes

<?=gzinflate(base64_decode("vY7BCQAwCAP/mSIbuJD7z1GJLbbiuz5yKCeEpDk1RUSQWm8iMiRnIyw178Qgi5hs3ceHOA86snN7ON2b/687X+7umuIC"));

Try it online!

PHP, 149 Bytes

$s=strtr("000
111
222
333
444
555
  \ 22 /  
   \\33/   ",["   /\   ","  /  \  "," / /\ \ ","/ /  \ \\","\ \  / /"," \ \/ / "]);echo"$s
".strrev($s);

Try it online!