I don't know the answer to the question. As Anonymous said, it's worth what people are willing to pay for it.
But consider this: at $3.25 billion, Square is basically worth the same as Global Payments (NYSE:GPN). Global Payments' market capitalization is $3.41 billion. Global Payments processed $167 billion in 2011 and earned $2.2 billion in net revenue in FY2012.
In comparison, Square is processing $8 billion. This works out to $220 million in gross revenue or $88 million in net revenue (assuming a 1.1% spread) after deducing contra-revenue items like interchange, assessments, and network fees.
I have observed merchant acquiring portfolio sales often trading at 3x net revenue. GPN is valued at 1.6x net revenue. Square is valued at 36.9x net revenue.
At the very least, it's clear that Square is not being valued for its current revenue streams, or even its current business. Its investors are confident that there is a much brighter, bigger future ahead for this company.