That's a big one. I'm trying without claiming to be near exhaustive.
The fundamental challenge for social sciences, as you rightly point out, is the absence of carrying out experimental designs on a large scale. Introducing a minimum wage in one half a country and not so in the other half would fail because of e.g. equity concerns (why would you advantage certain workers?) or external effects (firms moving from one region to the other, differential effects on trade).
On a small scale though, you might certainly be able to run experiments which might inform you on the large scale. An example would be random assignment in school to different pedagogical approaches. Tracking students over longer period of time might yield insights on how changing things might turn out on a larger scale.
Having said that, Economics, as widely understood by the public, deals with question on the large scale. The traditional approach to those question is to set up mathematical models which describe the behavior of economic actors. These models are not fundamentally different to what natural sciences refer to as laws (...of motion, of energy conservation). The difference is that social models do not (and cannot) claim their general validity and their reliability about the mechanism proposed. These models have been (and still are) derived from assumptions on individual (firms, households) or collective behavior (price mechanism). It is these assumptions which can be tested. They often have been tested with macro-scale regression analysis, such as explaining GDP growth by development aid to test effectiveness of those programs. While this is still done, at the research frontier such findings are by and large ignored.
Instead, Economics has undergone a so-called Credibility Revolution. In a nutshell, this entails the development and application of quasi-experimental methods which try to imitate the experimental ideal from natural sciences. If your question at hand is not feasible for a field experiment, you may look for settings where there is exogenous variation in a treatment variable that allows you identify its effect on a outcome variable of interest. There are numerous examples for this, also cited in the linked paper. One of the most famous ones is the comparison of increasing the minimum wage in one of two similar U.S. states. Exogenous variation may be geographic (compare two similar cities across a country border) or over time (a sudden reform that was not anticipated), to name just two. Nowadays, Empirical Economics tries to exploit these kind of variations with large-scale data on the micro level (the firm, the household, the individual) to come up with causal estimates on specific settings. If their findings can be confirmed by other settings, consensus on certain mechanism emerges, which might then be used to inform large scale theories.
To answer your question rightaway: Economists are not nearly as good as natural sciences regarding to the empirical basis of their theories, but they have greatly improved. On a more fundamental note, economic "laws" can never match the universality of natural science laws, as it's human behavior underneath.