Is it possible to combine covariates in a regression model? I'm not a statistician, so my question might have a trivial answer.
Let's say I want to calculate a logistic regression model, with 4-5 covariates. If I understand correctly, inclusion of covariates decrease the power of the model / require larger sample size.
So I thought that a possible solution would be to combine all covariates to one variable, for example to calculate a Z score for each variable, and then calculate a variable that is the distance of each combined variable (i.e. sqrt(x1^2 + x2^2 + ...) ). Of course then I will lose any information regarding the influence of each covariate to the model, but let's say I'm not interested in this information ... Is it correct to use such method? Has it been previously used?
 A: 
Of course then I will lose any information regarding the influence of each covariate to the model, but let's say I'm not interested in this information

You're saying additional samples decrease power, but if you aren't interested in interpretation then why are you interested in ameliorating this problem?  Additionally, your function is not one to one. This means there exist two points in your transform's domain which map to the same point in the image. Imagine there is a genuine relationship between predictors (say x1 increase the risk of outcome and x2 decreases the risk).  Your transform completely destroy information in the data because  (1,0) gets the same transformed value as (0,1) and the model can no longer detect the relationships.
In any case, what it appears you're doing is combining data to make a composite score of sorts.  Things like this are typically done via PCA which has the added benefit of giving you the composite score with maximum variance. If you insist on dimensionality reduction techniques, I would start there.
A: You are correct in general that "inclusion of covariates decreases the power of the model/requires larger sample size." Balancing the size of a data sample against the number of predictors and how best to include them in a model is part of the art of statistical analysis.
Combining predictors based on solid knowledge of the subject matter can be a good choice. You can see that in action in Chapter 14 of Frank Harrell's course notes, where several different types of blood pressure measurements are combined into one predictor, and electrocardiography results are combined with cardiovascular disease history for another, in a Cox proportional-hazards regression.
Your particular method certainly loses a lot of information, as comments and another answer note, and wouldn't be a wise choice in general. You might nevertheless consider principal-components regression (PCR) to be (distantly) related to your proposed approach, at least insofar as it similarly starts with standardizing all predictors to zero mean and unit standard deviation to put them in comparable scales. PCR reduces the dimension of the predictor space to something less than the number of original predictors by selecting linear combinations of predictors that contribute the most to overall variance.
The related method of ridge regression, which differentially penalizes the magnitudes of the coefficients of the predictors to achieve an effective reduction in the number of predictors, would probably be a good choice for a situation of this scale. The usual rule of thumb for an unpenalized logistic regression is to have about 15 cases in the minority class per predictor. So if you have 5 predictors of interest and fewer than 75 minority-class cases, ridge regression could provide a model that uses all of the available information in a principled way that avoids overfitting of your particular data sample.
In response to elaboration of OP
A later comment on the question indicates that the interest is in controlling for multiple covariates while evaluating the role of one primary independent variable. This paper illustrates use of ridge regression restricted to the potentially confounding covariates, and this post discusses several approaches including one close to that proposed in this question.
