Reddit mentions of The Metal Lathe (Build Your Own Metal Working Shop from Scrap)

Another one you may enjoy is the Build Your Own Metalworking Shop from Scrap series of books.

I would first like to say that I agree with everyone else here who recommends buying a used one and restoring it (or just keeping looking for a good deal, they're out there).

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That being said, there is a series of books available by David Gingery that has instructions on how to build basic metalworking machines from scrap. The first book in the series builds a foundry that enables you to cast aluminum, the second book builds a lathe, third a shaper, fourth a milling machine, etc. Seven books in total I think.

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If you do look into going this route, I'd strongly recommend looking into some of the more modern forms of sand casting, specifically the "lost foam" method which seems to be a lot easier to get consistent results from.

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Additionally, there are quite a few youtube series that build lathes either directly from Gingery's designs, or inspired by them. I was introduced to Gingery via the Makercise series and he, if I remember correctly, mostly follows Gingery's designs (he also covers lost foam casting as well)

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Note: if this series seems interesting to you, it would be cheaper to buy the hardcover book containing the whole series (or all 7 individual softcover books as a set) than it would be to buy each individually. However, if all you care about is the lathe, then just the first two should be enough to get you started (and you can always pick up the others later if you find yourself still interested in proceeding)

Might try checking out the series by David J. Gingery. It is 7 books that mostly build off of each other. You use metalworking techniques to build your own machining tools.

Charcoal Foundry

Metal Lathe

Metal Shaper

Milling Machine

Drill Press

Deluxe Accessories

Sheet Metal Brake

Otherwise there is always Instructables