OH BOY. This is actually what I did my PhD and Post doc doing!

We can! In fact we do it a lot.

As many others have pointed out, this is done through Solid Phase Peptide Synthesis (SPPS). Here we take a solid resin that looks just like plasticy sand, and amino acid (or two) at a time, we attach them using essentially fancy activated ester chemistry. We can do this pretty well up to ~80ish amino acids, with some new high flow reactors doing 120-150 amino acid proteins (<1% yield).

Basically the issue is the most simple reaction for adding a new amino acid is generally 3 chemical steps. 1) Deprotection of the resin; 2) Activating the incoming amino acid; 3) Coupling the amino acid. Then you repeat it over and over. If your efficiency of any of these steps is under 99.5%, you will have basically zero yield at longer peptides. To combat this, we've generated some amazing chemicals to make the active esters, and have begun using microwave SPPS to make the coupling happen very very fast and efficiently.

Now on to your real question. To make full length proteins, what we do is we do what's called Native Chemical Ligation. This is where we add on specialized termini to our peptides so that once we have a ~40-50 amino acid peptide, we can mix them together and boom, we have a longer peptide. Repeat this a handful of times and you can make longer proteins.

This has been done to make a lot of histone proteins, including histone H1, which is 212 amino acids that included a number of post-translational modifications. I think this had an 8% overall yield? I forget

Now that we know what we're doing, it still takes ~1-2 years for a post-doc to work out all of the kinks for a longer protein before they can make enough to do research with it. For larger amounts of material it is probably not worth it.