I want to take a crack at building my own bike. I found a lovely old Peugeot road bike frame, free, but it's 25" and I'm 6 feet tall; too short for the bike, at least according to the size charts online.

Can I make it work by putting the seat all the way down, or should I just not waste another second on this frame?

I'd pass .

I'm 6'1", 34" inseam.
23" steel frame is my limit.
YMMV

It's not just the saddle height. The top tube might be too long, requiring a shorter stem that could affect handling.

My '89 steel road bike is technically the right size but due to an old neck injury it felt too stretched out on the original stem. I replaced it with a shorter stem which is much more comfortable. But I needed a couple of cautious rides to adapt to the slightly different handling, especially on fast downhills and turns. It's not twitchy, just more responsive. Feels fine now.

And if you install tires that are larger than were common way back when, it'll make the bike taller. One of my hybrid bikes is technically the right size, but was originally spec'd with 700x32 tires. After I put on 700x42 tires the bike felt considerably taller. The bike still fits. But I need to be more cautious setting a foot down at stops. Next time I build up a city bike from an old steel diamond frame bike, I'll go a size smaller to accommodate my preference for bigger tires.

Sure you could make it work, but why?

I am 6'5 and ride 25" Schwinn frames/64cm level top tube frames. I am not sure why you would want to go thru the process of building up a frame only to have to make obvious and perhaps extreme concessions to make it fit.

There are a ton of quality frames for $50-175 available that will fit properly. Check Ebay, local/regional Craigslist, and the BF C&V forum linked below.
Total guess, but something 56-60 would fit you best, correct?

Check out this subforum- https://www.bikeforums.net/classic-vintage-sales/
it is very active and i have yet to be disappointed in buying or selling on it.

SR Maxima frameset as an example that just sold recently.
Vintage frames for sale options to also repaint.
FS: Bikes/Frames/Wheels - Southern California the Merican is really cool

Haha, you're right. I dunno, I'm in Canada so maybe that's making it harder to find a good frame, but I'm just having a really hard time finding something nice, and shipping from the US would open up my options but is a PITA.

I'll keep searching!

You can't base anything on the seat tube size alone. You need to know what the reach is going to end up as well as whether you can have an appropriate saddle height. I wouldn't want to have my saddle all the way down. For one, you don't want to have the top tube interferring with your treasured personal items when you straddle the bike and for another it means you have nowhere to go but up when you need to change the height for a different saddle design.

Reach is the trickier to consider because it involves not only the length of the top tube, but the angles of the head tube and seat tube as well as the stem length and handlebar specs.

Honestly, height-based bike sizing makes as much as basing your pants size off your height... It all depends on your bodys proportions, especially your legs, aka cycling inseam (crotch to floor, not what your pants say). I'm 6'1", but my cycling inseam is 36". I have a 25" frame bike and absolutely love it. Being an older frame also, the top tube is short for the seat tube (how they built the old bikes) which suits me just fine! I'll usually get a frame in the 60-64cm area. A 58m/23" gets me a very slammed position since there is something like 8" or more of seatpost exposed!

Not knowing your proportions, I'd suggest trying the Competitive Cyclist fit calculator. It's certainly not perfect, but will get you in the ballpark for the size of frame that should fit best. Or, just straddle the bike. If you can stand with both feet flat and still lift the front wheel off the ground a bit without endangering your anatomy then go for it.