Square taper spindle won't fit into the bottom bracket I'm making a road bike with random components I have in my garage.
Everything works ok except for the bottom bracket which is a square taper 115mm long and it is still spinning free even if I turn the adjustable cup all the way in.
Whats wrong? Do the spindles have different diameters?
Since you are fitting Italian cups I assume the bottom bracket shell is 70mm wide according to Italian standard - see Sheldon Brown's Threaded Bicycle Bottom Bracket Crib Sheet.
This shell accepts the spindles with race distance of 55mm - see Bottom Bracket Size Database, section Accuracy of the Sheldon Brown's page again.

As you mentioned you are using random parts so there is a chance that the spindle is made for British ISO where the bottom bracket shell is 68mm wide and the distance between the spindle races is 52mm. This would explain why you cannot set the correct play in the BB bearings.

If I may suggest a solution - I'd use a bottom bracket package with cartridge bearings - for about €20 you get a maintenance-free setup and you get rid of the bearing preload setting problem. I don't know your location but in my case I'd source the parts here (no affiliation whatsoever).



EDIT: Measuring the spindle width.

Choosing the correct spindle width if you don't know the exact parameters of your parts is often a trial-and-error process. You can get some approximate measurements to start with in the following way:  

Have a look at the picture attached. Measure the current spindle width (the distance between the ends of the square tapers).

Mount the cranks (blue) to they come in the same direction and measure the inner distance between them (measurement A). Mark the place on the chainstays (green) where the cranks could engage them and measure the outside distance (measurement B). 

The rule of thumb (what worked for me) is that A must be greater than B by about 6 mm (3 mm on each side) to prevent the cranks from hitting the frame should any of those flex. 

Now, knowing your A, B and the current spindle width figure out what your minimum spindle width should be. Most likely you need a wider spindle than you now have.

Then, insert your spindle into your bottom bracket (with the bearing), mount the crankset and push the spindle to the drive side like it was tightened. Check whether the chainrings do not hit the chainstay and whether they would not be hitting it if the spindle was of the correct width (remove the chainrings and replace them with the cardboard mock-up if they currently do). Also check whether the middle chainring allows for a straight chainline with the cassette's middle sprocket. Or check the distance against the middle of the seat tube. I can't recall the typical road bike distance at the moment but you can search the internet for it.

Remember that the spindles are not symmetrical - the side with greater distance from the bearing race goes on the drive side.