Q:

Is economics a science? 

In class, I made the argument that economics is not science, because it cannot undertake repeatable experiments.  Someone rebutted: this would mean that I am ignore an emerging body of work, some by prominent Nobel Prize winners, that is transforming economics into a field firmly grounded in fact.  
How would you justify whether economics is scientific or not? 

A:

Welcome to the demarcation problem of science.  What is this thing called science?  In lower levels of education, one is often given the impression that 'science', whatever that may be, exists as a monolithic entity.  There is no sufficiency and necessity definition of what science is.  It's better to say 'sciences' or 'scientific' when speculating as to this question.  While physics has often been touted as a role-model of the sciences with its focus on measurement and manipulation of variables, in fact, each of what are often seen as fundamental sciences, physics, chemistry, biology, psychology, and sociology, have different methodologies and are roughly lumped into the natural sciences (the first three) and the social sciences (the last two), though newer tools for exploring mind and data have "hardened" the "softer" sciences.  Today, one can use fMRIs, for instance, to peer directly into the operation of the brain when trying to diagnose abnormal psychological states. 
Wikipedia isn't inaccurate in it's definition of economics: 

Economics (/ɛkəˈnɒmɪks, iːkə-/) is the social science that studies the production, distribution, and consumption of goods and services.  

So, economics is considered a science, but how is this so?  For starters, your premise that variables can't be manipulated is wrong.  Behavioral economics, for instance,  uses some hard science:

Behavioral models typically integrate insights from psychology, neuroscience and microeconomic theory. 

In fact, Daniel Kahneman and Amos Tversky with their prospect theory have helped to reshape our notion of economics and make it even more scientific than ever by killing the idea sometimes known as Homo economicus, the rational economic agent.  Behavorial economists often manipulate variables in line with their psychological training. 
How then is it a science?  Well, there's more to science than manipulating variables.  For instance, it's possible to draw causal inferences in nonexperimental research and sophisticated statistical modeling can be used for prediction.  Statistical or probabilistic explanation has value too.  Operational definitions are used to define abstractions, and theories still have mathematical and logical aspects that can be explored rationally.  In general, like the hard sciences, the soft sciences can use deduction, induction, and abduction to eke out certainty from the world around us.  While complex phenomena do not reduce as easily as simple physics experiments, nonetheless they are subject and object of scientific discourse. 
REFERENCES
Blackwell's Companion to the Philosophy of Science
Blaclock, Hubert M. Causal Inferences in Nonexperimental Research
Kahneman, Daniel.  Thinking, Fast and Slow

A:

What is Science? 
The Popperian view of science is that a claim is "scientific" if it can be falsified.   Science cannot prove that a hypothesis is true, only that it is manifestly false.   If economics can make falsifiable claims, then I think it is justified to say such claims are "scientific", at least on some level (the degree of repeatability is certainly a basis for ascribing a non-binary "scientific" label). 
Natural Experiment
Thus, even if one cannot control the variables, it is still possible to "do science", as another answer noted for astronomy (at least until we master astro-engineering ;).   An example of a "scientific astronomical claim" is that inter-galactic cosmic rays cannot exceed 5x10^19 eV, known as the "GZK limit".   Since cosmic ray detectors can measure the energy and direction of cosmic rays, we only need to plot histograms of energy vs. source to determine whether the claim is falsified or not. 
When a claim can be tested merely by accumulating observations under both the experimental and control groups without intervention by experimenters, you have what is called a "natural experiment".   In the cosmic ray case, the experimental group are cosmic rays emanating from a known galaxy, while the control group are cosmic rays not emanating from a known galaxy (i.e., coming from a source within the Milky Way).   Even though we cannot control, let alone cause, the emission of cosmic rays from any stellar source, we can still "conduct" the natural experiment just by watching the sky for a while. 
As it happens, such experiments also occur in economics.   As you may know, there is currently debate about the effects of a high minimum wage.   It would be unethical and really, unthinkable, to conduct a randomized controlled trial (RCT) in which half of the citizens in a city got the high minimum wage, while the other half got the federal wage.   However, it is possible to infer the effects (and even make falsifiable predictions) by observing before-and-after economic effects in a city which did implement a high wage and comparing it to a similar city which did not.   Economists from the University of Washington did exactly that.   While this study was primarily observational, it would have been just as easy to start with one of the partisan claims, such as: "Average restaurant prices will increase by more than 5%" or "Unemployment levels will remain the same, to within the margin of error."   These are clearly as falsifiable as the claim: "No high-energy cosmic rays can reach us from another galaxy." 
Policy
Another answer observed that economics and politics are intimately entwined.   We would like any policy with an economic impact to be informed by the best knowledge we have available.   Thus, whenever Congress proposes a new law, the CBO is asked to create forecasts of the costs and impact of the proposed law.   That forecast is essentially a falsifiable prediction, based on economic models which serve as the "economic theory" on which Congress debates.   Thus, as a democracy, we act as if economics is a "science" whenever we weigh the CBO forecast for the cost of an entitlement program like the ACA or a tax break like the Tax Cuts and Jobs Act. 
In some cases, economists act like experimenters when they propose policy with an explicitly economic effect.   For instance, consider the "Cash for Clunkers" program proposed by economists and signed into law by Obama.   Of course, the economists had no control over who received the rebate and who didn't, so again, RCT is infeasible for pretty much any national economic effect.   Apparently, no economists dared to make quantified claims about the results before the program went into effect, so this isn't quite as scientific as other programs. 
A much better example is the American Recovery and Reinvestment Act (ARRA, but you can call it the "Obama Keynesian Stimulus Package").   As expected, the CBO provided a forecast for the program cost, as well as impacts on GDP and employment.   The very notion of government economic stimulus in the first place is due to Keynes and can be considered a fairly scientific theory, even if it is not as precise as one may be used to from the hard sciences. 
Conclusion
Of course, the way results are interpreted is open for debate, and much controversy surrounding the UW minimum wage study revolved around whether they had an appropriate control city for comparison.   These are the weaknesses of not having a fully experimenter-defined control group.   But, not being able to control every aspect of the universe should not prevent us from attempting to pose and test scientific claims about how those phenomena work.   Is economics as scientific as Newtonian mechanics?   No, it is not.   But science is a continuum, and I think it can make sufficiently detailed falsifiable claims to put it squarely on the "science" side of the continuum. 

A:

Wikipedia says

Natural science is concerned with the description, prediction, and understanding of natural phenomena based on empirical evidence from observation and experimentation.  It can be divided into two main branches: life science (or biological science) and physical science. 
Social science is concerned with society and the relationships among individuals within a society.  It has many branches that include, but are not limited to, anthropology, archaeology, communication studies, economics, history, human geography, jurisprudence, linguistics, political science, psychology, public health, and sociology.  Social scientists may adopt various philosophical theories to study individuals and society.  For example, positivist social scientists use methods resembling those of the natural sciences as tools for understanding society, and so define science in its stricter modern sense. 

Economics is a "social science".  Should we also categorise it under the bare label of "science" as well? 
I think it would make sense to do so.  Most social sciences are based on the first process of the natural sciences: observation.  While observation would not suffice for most natural sciences, some like astronomy are arguably predominantly observation based, as humanity is not currently capable of affecting star formation etc. 
Experimentation is tricky for the social sciences because of the ethical questions around using human subjects.  But with willing volunteers experiments can be made.  For example, some people create their own languages, called conlangs, in order to test theories of language, and then teach them to other people. 
Economics clearly has a strong observational and theoretical history, but there are also times in which people have experimented with economic ideas.  I would say that Basic Income trials would count as economic experiments for example.  With observation, analysis and theorising, and experimentation, I think economics should be considered a science.