Is it possible to figure out the public key from encrypted text? Suppose Alice sends messages to Bob by encrypting the messages with Bob's public key.

Eve knows that the data is encrypted using RSA, but does not know the public key. Can Eve figure out the public key just by observing the encrypted messages?

And if so, approximately how much data would it take for Eve to discover the public key?

(assume that the public key used is the same throughout, and that Eve can potentially grab as many encrypted messages as she desires)
Actually, if Eve does have access to Alice's plaintext messages (and Alice doesn't do random padding as she should), then yes, Eve does have a chance, even with only two messages.

Suppose she guesses $e$ (the public exponent), and that it's not a not huge value (say, the common value 65537).

Then, for every plaintext/ciphertext pair $P_i$, $C_i$, she knows that (if $N$ is the unknown modulus):

$P_i ^ e = C_i \mod N$

or, in other words:

$P_i ^ e - C_i = k_iN$

for some integer $k_i$.

So, if she takes two plaintext/ciphertext pairs, and computes the value:

$gcd( P_1 ^ e - C_1, P_2 ^ e - C_2 )$

that is extremely likely to be the value $N$, possibly multiplied by a small integer.  Now, this gcd operation might be on operands of possibly a few hundreds of millions of bits long; not something you want to do every day, but it certainly in the realm of possibility if you're a bit patient.

Now, if Alice does do random padding before doing her RSA encryption, well, Eve doesn't have access to the values $P_i$, so this observation doesn't help her.