I am not expecting a "realistic" answer to my question, since it is based on an impossible scenario. What I'm waiting for is a theoretic explanation/interpretation so that I can sleep at night :)
Let's take n reals from the interval (0; 1), evenly distributed. What is the probability of the sum of squares being less than 1? With a geometric approach it is relatively easy to see that the solution is $P\left(\sum_{i=1}^nx_i^2<1\right)={\pi^{\frac n 2}\over 2^n\cdot{\frac n 2}!}=\frac {volume \;of\;hypersphere}{volume\;of\;hypercube}$ This solution works fine for all nonnegative integer n (for odd n we compute the factorial using the $\Gamma$ function).
But what if n could be something else, namely a real from (0; 1)? In that case the resulting probability is greater than 1, (properly) indicating that something went wrong. Is there a way to make sense of this result?