Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom?
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1. **Determine the relationship between the side length of the hexagon and the radius of the spheres:**
   - Let the side length of the hexagonal bottom be \( a \).
   - The radius \( r \) of each sphere is given by \( r = \frac{\sqrt{3}}{2}a \) because the diameter of the sphere is equal to the distance between opposite sides of the hexagon, which is \( \sqrt{3}a \).

2. **Calculate the area of the hexagonal bottom:**
   - The area \( A \) of a regular hexagon with side length \( a \) is given by:
     \[
     A = \frac{3\sqrt{3}}{2}a^2
     \]

3. **Determine the height of the glass:**
   - Since there are three spheres stacked vertically and each sphere has a diameter of \( 2r \), the total height \( h \) of the glass is:
     \[
     h = 3 \times 2r = 6r = 6 \times \frac{\sqrt{3}}{2}a = 3\sqrt{3}a
     \]

4. **Calculate the volume of the glass:**
   - The volume \( V \) of the glass is the product of the area of the base and the height:
     \[
     V = A \times h = \frac{3\sqrt{3}}{2}a^2 \times 3\sqrt{3}a = \frac{3\sqrt{3}}{2} \times 3\sqrt{3}a^3 = \frac{9 \times 3 \times 3}{2}a^3 = \frac{81}{2}a^3
     \]
   - Given that the volume of the glass is \( 108 \) cm\(^3\):
     \[
     \frac{81}{2}a^3 = 108
     \]

5. **Solve for \( a \):**
   - Simplify the equation:
     \[
     81a^3 = 216
     \]
     \[
     a^3 = \frac{216}{81} = \frac{24}{9} = \frac{8}{3}
     \]
     \[
     a = \sqrt[3]{\frac{8}{3}}
     \]
     \[
     a = \frac{\sqrt[3]{8}}{\sqrt[3]{3}} = \frac{2}{\sqrt[3]{3}}
     \]

The final answer is \( \boxed{ \frac{2}{\sqrt[3]{3}} } \)