If you fix $N$ and group elements $g_1$, ..., $g_N \in G$, then your question becomes closely related to tilings of groups. Specifically, in Chapter 2, section 2 of "[Entropy and Isomorphism Theorems for Actions of Amenable Groups][1]," Ornstein and Weiss prove:

Let $G$ be a countable group acting freely and measure preservingly on a standard probability space $(X, \mu)$. Fix a finite set $T \subseteq G$. If for every $\epsilon > 0$ there is a measurable set $U \subseteq X$ such that the $T$-translates of $U$ are disjoint and $\mu(T \cdot U) > 1 - \epsilon$, then $T$ tiles $G$ in the sense that there is a set of centers $C \subseteq G$ such that the sets $Tc$ ($c \in C$) partition $G$.

They also prove that if $G$ is amenable then the converse holds. So if $G$ is amenable and $T$ is a tile for $G$, then for every free probability measure preserving action of $G$ and every $\epsilon > 0$ there is a $(\epsilon, |T|)$-fundamentalish domain. Thus a natural question is: which amenable groups admit arbitrarily large finite tiles?

Weiss called a group $G$ MT (mono-tileable) if for every finite set $F \subseteq G$ there is a finite tile $T \subseteq G$ containing $F$. In "[Monotileable amenable groups][2]," Weiss proved that all solvable groups and all residually finite groups are MT. In "[Elementary Amenable Groups][3]," Chou proved that all elementary amenable groups and all free products of non-trivial groups are MT. So in particular, your question has a positive answer whenever the group $G$ is elementary amenable. A stronger tiling condition, called ccc, is studied in chapter 4 of "[Groups Colorings and Bernoulli Subflows][4]" (this paper is in preparation). Weaker properties of poly-MT and poly-ccc are studied in "[Burnside's Problem, spanning trees, and tilings][5]." To the best of my knowledge these are the only papers which study tilings of countable groups.


  [1]: http://www.springerlink.com/content/7p0r815012335131/
  [2]: http://books.google.com/books?id=NDJ0rRuSMScC&pg=PA257&lpg=PA257&dq=monotileable+amenable+groups&source=bl&ots=DpqekXyi_H&sig=ShnrG6LH5nqIqT1QKGWVcxJFXlM&hl=en&ei=aovBTs3_D8W02wWAs-GxBQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBoQ6AEwAA#v=onepage&q=monotileable%2520amenable%2520groups&f=false
  [3]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ijm/1256047608
  [4]: http://www-personal.umich.edu/~bseward/Files/Group%20Colorings%20and%20Bernoulli%20Subflows.pdf
  [5]: http://www-personal.umich.edu/~bseward/Files/Burnside%27s%20Problem,%20Spanning%20Trees,%20and%20Tilings.pdf