To answer the physics/engineering question - no, there's nothing really "stopping" us from launching orbital data centers. You'll note that most responses so far focus on the economics, and not the question of whether or not it's possible to do in the first place.
So, there's only one question that matters - is launching and operating orbital data centers cheaper than building and running a terrestrial data center?
There are three financial aspects of "building" a data center- the initial capital expenditure, the recurring operational expenditure, and the revenue it generates. The asset comparison is between launch cost + computers + satellite vs. building + computers.
Our first comparison is the cost of a rocket launch vs building a building. Here, the big technology enabler is SpaceX. SpaceX has been driving down launch costs for years, and Starlink is proof that significant reduction of launch costs can create new markets with fairly respectable profit margins. If this trend continues, then the capex math of launch vs build will continue to shift in favor of orbital data centers.
The second comparison is between building and operating satellites compared to outfitting and operating data centers. Here, it's a lot less concrete. Orbital and terrestrial data centers each have their pros and cons. For satellites, you have better solar panel efficiency, manufacturing economies of scale, but radiation-only cooling, space-to-Earth data transfer, and no maintenance access, requiring higher redundancy, rad hardening, and the like. On the ground, we have, well, many more options.
But it's not immediately obvious which of the two is better when it comes to capex and opex combined. It's clear which is harder to do, but it's not clear which is cheaper to do.
All of this pales in comparison to revenue. Because everyone is so insanely AI-crazy right now and starving for more compute, the potential revenue can justify a relatively high cost (and high risk) business. Like someone else mentioned, orbital data centers don't really make sense if you're launching an ordinary data center with ordinary revenue numbers.
There's a fourth dimension here, which is time to scale. Regulations, permits, and all the other challenges of construction can slow down your deployment significantly. None of that is required in space. How significant this is, you'd have to ask someone who understands construction better than I, but I suspect this could be a significant reason for the attraction to orbital data centers.
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Nuances involving orbits, rocket payload capability and availability, and more have been omitted for simplicity. I don't have the numbers - the above is just to highlight the relevant principles.