I recently did a build with a 80's (I believe) Peugeot road frame and turned it into a singlespeed commuter. After installing some 700c wheels and some tires 23 up front 25 in the rear, I realized that there was over an inch of clearance left both up front and back! I am confused on what brake caliber I can get to reach the wheels and I'm also wondering if this is some special kind of bike that had big wheels or something

MAFAC Racers. They reach, they work. And no, it was nothing special - a lot of bikes back then actually had room for wide-ish tires and fenders.

SP
Bend, OR

I agree with bobbycorno. Mafac Racers will work well AND they're French. You could also use any of the Tektro extra long reach calipers. I suspect that your bike was originally equipped with 27" wheels and may have been equipped with Weinmann centerpulls. Since you have the clearance, you will notice a much more comfortable and forgiving ride with 28-32mm tires.

CLB aslo made a few sidepull models with long reach, but I suspect they will not brake as well as Mafac CPs as CLBs tend to be on the weight weenie/wimpy side with most of their calipers....but just in case you want another French option, you might consider them. Otherwise many 80's and 70's Peugeots were also outfitted with Weinmann brakes they also have many models that have long reach....but they are certainly not French made.....

Chombi

I was always instructed to put the wider tire up front. Better handling that way. And the wider tire absorbs the bumps better. Rear wheel just sort of trails along.

Yeah, but... The rear wheel also carries more load and gets more stress as a result (and is inherently weaker with its off-center construction). The wider tire would afford more protection for the rim, and wear better. And "better handling"? Not IME - a wider tire in the back tends to stabilize the steering. No clue why, but that's how it's worked for me.

SP
Bend, OR

Maybe a wider tire at the rear puts a little bit more rolling resistance/drag at the back giving the effect of what fletching does to an arrow.

Chombi

I tend to underfill my front tire a little bit (only for polo) with the idea of better handling in mind, it seems to just hold onto the pavement a little better when my bars are parallel with my frame, but I really think that's the only application I can think of for something like that where the cons dont outweigh the pros.

Mafac Racer brakes work great. They take a long time to set up right. Be sure not to make the yoke cable long. When it's long, it decreases your leverage.

Mafac Racer brakes are a real PIA to set...Rivendale bike has a set of dual pivot calipers for older nutted style brakes. I'd go with the dual pivots...