First, lets take a second to understand HOW things are calculated. Some problems in calculus have what are called exact solutions. You can use methods to figure out an exact solution for EVERY POSSIBLE VARIABLE. Most problems in practice, have exact solutions that are so difficult to determine it's just not practical. A branch of mathematics called Numerical Methods was developed concurrently with calculus to figure out approximate solutions was developed. In engineering you rarely need an exact answer for something, and usually knowing something within a few percent is good enough.

Many of these methods require you to do many iterations of the same problem with new inputs and can take up vast amounts of time. A Computer was the job title of someone who's job it was to solve these problems. The Manhattan project, for instance, employed many people to simply compute answers to problems. Mechanical and modern computers both use the same Numerical Methods techniques to do the exact same thing, only much faster.

Painstakingly. There also used to be books with mathematical tables: e.g. a whole book full of logarithms. So the publisher had done the painstaking work of calculating all the logarithms for you, and you just had to be able to use some basic transformations to convert your inputs into the rows and columns of the tables.

Many approximation methods were developed, such as Newton's method, or Taylor polynomials, which used easier functions such as lines and quadratics to approximate more complex quantities such as sine and cosine to any desired level of precision.

Note that irrational numbers aren't actually that big of a deal computationally. Only 39 digits of Pi are needed to calculate the size of the universe within the size of one atom. Of course, for less precise things (which is virtually everything), only a few digits will do for designs, simply because any mathematical approximation of, say, a beam has a significant amount of error that's taken care of through a factor of safety.

Besides the tables used that are already mentioned, a big one is the slide rule. It's a physical device which is used to multiply or divide. They were invented in the 17th century, and were ubiquitous in the 20th until the calculator replaced it. While not as precise as a calculator, it doesn't make much of a difference for most purposes, and if you know what you're doing it's just as quick. For some things, like unit conversions, a slide rule specifically designed for the purpose may even be faster. Watches designed for flying often have circular slide rules, which have common unit conversions labelled on the band.