What is it like doing research in macroeconomics? Do people actually travel to certain countries to do this, or is it solely just a lot of number crunching, with vast data sets, in front of a computer?

It really depends on the type of research you are doing. For the most part though it is more number crunching than traveling. This is especially the cass in the modern era. Even if you are trying to study historical texts can request them to be scanned. Most importantly though, current macroeconomic research must be verified by data. This is because the rise of data science has led economists to believe they can model the macroeconomy.

The best proof of this is that econometrics is now a requirement for most economics majors. They are taught to use STATA. Many argue these students are being taught to rely to heavily on statistical techniques that can't truly model the economy though.