Entropy is defined for distributions and it's problematic to apply it to unbounded bit string. Normally if you have k possible event types, you observe n events, with p'th event occurring $i_p$ times. You define your distribution to be $(i_1/n,\ldots,i_k/n)$ and compute the entropy of that. But how do you define "events" when you get a string of length n? You could define your events to be "observed bit 1" and "observed bit 0", or you could have events of the form "observed string x" where x is some string of length n. In latter case, your entropy is going to be 0.
In practice, there's no universal test for stream randomness, instead there's a series of tests, and if your stream tries k of the best tests and passes them all, we can be reasonably sure it's random...until someone invents k+1'st test that breaks it.
Here's what Knuth says about it in "Art of Computer Algorithms, Vol 2"
"If a sequence behaves randomly with respect to tests T1 ,T2 , ..., Tn, we cannot be sure in general that it will not be a miserable failure when it is subjected to a further test T(n+1). Yet each test gives us more and more confidence in the randomness of the sequence. In practice, we apply about half a dozen different kinds of statistical tests to a sequence, and if it passes them satisfactorily we consider it to be random - it is then presumed innocent until proven guilty."
I'd recommend reading Knuth's "Art of Computer Algorithms" section 3.1 for general introduction to pseudorandomness and 3.3 on statistical tests for streams.