That’s called solid phase peptide synthesis. You start with a single amino acid linked to a solid resin through a cleavable linker. The peptide is built up with amino acids that are selectively protected on the amine (and usually any reactive side chain functional groups) and have a free carboxylic acid. The acid is ‘activated’ into a more reactive form and added to the resin where the growing chain has a free amine to react with it. Then the reactants are washed off and the amine protecting group is chemically removed to expose a new amine and the cycle repeats. When the full peptide has been made there is a global deprotection process that also cleaves the peptide from the resin. It usually needs HPLC purification to separate it from peptides that may have missed a coupling or two.

The major power of this method is that you can readily introduce all sorts of unnatural amino acids or even entirely different kinds of chemical functionalities (e.g. esters instead of amides). While there are ways to do that biochemically, there’s much more flexibility with solid phase synthesis.

As others have noted, there are some limitations to the process. The most important is scale - because it’s being built on a resin you’re limited by the number of sites where a peptide can start from, so you’re generally going to get milligrams to tens of milligrams out. Second, there can be complications depending on how favorable it is for the growing peptide to fold in on itself, which can happen even using strong solvents like DMF. That can keep the end of the peptide from efficiently reacting, so you’ll end up with errors.

With all of that said, it’s relatively trivial to make 20-mers with automated synthesis. Once you get out to the 60-mer range it becomes challenging but not impossible. Much longer than that and you’re probably better off with biochemical synthesis, either in cells or a cell-free extract.