How can I estimate a confidence interval for experimental results with only one run?

Its seems to me as a rather complex experiment that requires serious probabilistic modeling rather than a simple estimate of confidence intervals.

Simplistic view
Confidence intervals are a tool of frequentist statistics which rely on making multiple measurements. For example, given a set of repeated measurements ${x_1, x_2, ..., x_n}$ and assuming Gaussian statistics, the mean and the variance can be estimated as $$ hat{mu} = frac{1}{n}sum_{i=1}^n x_i, hat{sigma^2} = frac{1}{n-1}sum_{i=1}^n(x_i - hat{mu})^2, $$ where the hats indicate that these are estimators rather then the actual values. With only one measurement the estimate of mean is just the measured value, whereas the unbiased estimate variance is non-existent, since $n-1=0$. Even if one agrees to use a biased estimate of variance (with $n$ instead of $n-1$), with only one measurement it is identically equal to zero. Without the variance one cannot find the confidence intervals, which is probably the basis for this question.

Maximum likelihood approach
However, when multiple quantities ${q_1, q_2, ..., q_n}$ are measured, which are all dependent on a smaller set of parameters $theta$, one can construct the likelihood of observing this set of quantities $$ P(q_1, q_2, ..., q_n|theta) $$ Maximizing this likelihood gives the estimator for the parameters, whereas calculating Fisher information gives the estimate of the variance, which, in principle, allows calculating confidence intervals (although statisticians might disagree that calculating confidence intervals is appropriate in this case).

Bayesian approach
A problem with a single measurement is probably more naturally treated in the Bayesian framework, using the Bayes theorem to define the probability distribution of the confidence intervals, given the observed set of diverse measurements $$ P(theta|q_1, q_2, ..., q_n) propto P(q_1, q_2, ..., q_n|theta)P(theta), $$ where the prior probability, $P(theta)$ reflects the initial knowledge about the parameter variation, e.g., due to the known error/precision of the equipment. One can then trivially estimate credibility intervals, which are the equivalent of the frequentist's confidence intervals for the Bayesian case (and in fact are easier to interpret). Let me note however that using the Bayesian versus the likelihood framework is a matter of convenience here.

Finally, you might get more answers, if you post a question in the statistics forum (Cross Validated), although your description of the experimental details might have to be streamlined/simplified.

Estimation of confidence intervals for quantities from a single complex situation requires that you have a reliable statistical model of the whole situation relating all the data that you have form the various experiments done. Under suitable assumptions, the unknown parameters and error covariance matrices entering the model can be estimated from the avaliable data, which gives an assessment of the accuracy of a single experimental result form the totality of measured data.

The proptoype of such an estimation method is the restricted maximum likelihood (REML) procedure, about which there is a large literature and overwhelming positive evidence for its usefulness. For example, REML is the basis of the commercial methods for animal breeding, where the quantities of interest are breeding values of key animals to be used for the planning of breeding and the model relations come from the Mendelian laws. See, e.g., my paper

• A. Neumaier and E. Groeneveld, Restricted maximum likelihood estimation of covariances in sparse linear models, Genet. Sel. Evol. 30 (1998), 3-26.

REML is most developed for models linear in the parameters (but nonlinear in the covariance components). Extensions to nonlinear models are also possible but are much less explored.