Modern bike shops in the USA call those tires "Tubulars", otherwise they go by "Tubs", "Singles", "Sew-ups" or "F$^* these!" if you're installing them yourself and you think it will be easy or fast. Tubulars do ride amazing.
Tubular tires are definitely still in production, and unless you plan on doing actual hard cornering on that bike, you won't need to do the level of work that a UCI Cyclocross mechanic does to glue the tires on and ride the bike. The last shop I worked in Portland OR would install tubulars as a service, labor was $125 per tire + $100 per wheel if they had to scrape old glue, plus around $10 per wheel in glue, on top of a fairly expensive tire ($100-150 each) on wheelset(s) that rarely cost less than $1-3k per set. Most people that paid that were well off racers who were either good enough to think it would give them an advantage, or rich enough to not care about the cost.
If you just want to ride the bike and feel that it mostly needs new tires and some love for the chain, I'd just see if you can get that old chain going, and then try the laborious, dirty, glue fumes heavy process of stretching and gluing some modern tubulars yourself!
If the outer diameter of your wheel is roughly similar to a modern 700c Tubular (I would guess that it probably is) and you have a local bike shop you like, Challenge makes a tire called the "Strada Biancha Pro 700 x 40" that your shop could order from QBP, and would probably clear your frame based on your pictures (do some measurements though before taking my word for it).
That bike looks cool, especially the handlebars, and what looks like Industry9 style straight pull spokes (though I couldn't really tell that one from the pictures).