Choosing finite field size in Shamir's Secret Sharing Scheme
Actually, you can do Shamir Secret Sharing over any finite field $GF(p^k)$, for any prime $p$ and any integer $k$.  If $k=1$, you have the $GF(p)$ field you mentioned; however it works on extension fields as well. We often pick $p=2$ and $k$ a multiple of 8; this makes everything nice even number of bytes (at the cost of doing our calculations in $GF(2^k)$).  Of course, this is a choice of convenience, you can make your own choice.

As for the size of $p^k$, well, $p^k &gt; n$ is a hard requirement; you can't give out more shares than nonzero indicies available.  As for $p^k &gt; S$, well, that's not actually a hard requirement; you can split up $S$ into several pieces (each piece no larger than $p^k$), and share each piece separately.  For example, if the secret is a 128-bit AES key, and you're using $p = 257$, you'd split up the key into 16 pieces, each from 0-255, and generate 16 independent polynomials, and give each share holder a point on each of those 16 polynomials.  Just remember: when generating the polynomials, the polynomial you pick must be selected independently of all the other pieces; otherwise you lose the security of the system).

As for not picking $p$ (or $p^k$) too large, well, Wikipedia is wrong there; Secret Sharing is informationally secure (assuming a randomly chosen polynomial); if someone has fewer than the threshold number of shares, then they learn nothing about the secret, and that holds no matter how large we make $p^k$.  I suspect someone took the discussion about Shamir Secret Sharing over the integers (which doesn't work, in part because the integers don't form a field, as there are elements without multiplicative inverses, and in part because it's impossible to select a random polynomial with a uniform distribution over the integers), and extrapolated the problems there to finite fields (which doesn't have those issues).