How do I keep smallest gear on my crank shaft from contacting frame I have an old Trek 8000 ZX that I am fixing up. I had to replace the bottom bracket and I believe I successfully did that. The bike did not have a crank set to work with so I salvaged one from a pile of other old bike scraps. 

My problem is when I attached the crank arm and tried to spin the pedal, the smallest gear just rubs against the frame.  It appears that the crank is too close the frame. It is just making contact 5mm would address the issue.I want to know what my options are. My initial guess is to find a crank set with smaller gearing and hoping to find one that fits. The other idea is try and get a spacer but I am not sure how safe that would be. 

There are number of bikes available that I could steal parts from but I would like some guidance on what I should be looking for in terms of size. I am not sure of the age of the bike, but I am pretty sure the frame size is 21". The color is yellow if that helps. 

I am sorry if some of my terminology is wrong/crude, I am new to bike repair/salvage (Having a blast though). 

Thanks in advance.
Welcome to BB standards hell...... you are about to learn far more than you wanted to about bicycle gearing cranks and chain rings.....The only spec I could find on these was the 1999 model, which used a Shimano BB-UN52, 113mm spindle BB Shell Width   73mm - its pretty common and typical of that era MTB. 

Your idea of another crank set is a possible solution, that might be easier if you have a supply of donor bikes, is a larger small chain ring. 

Its likely though you need a longer bottom bracket spindle, as a short spindle will not only cause this to be a problem, but will affect the chain line. With incorrect chain-line you may have problems with getting gear shifts to work smoothly and reliably, and have problems with chain drops.  Read up about chain line Here (as always, Sheldon Brown)

Presuming its square taper BB, they come in a range of lengths. You should be able to measure your current length, and might find a longer one on your donor bikes. Hopefully the one you have now is around 106mm, meaning all you need to do is source the correct 113mm sized one.