Normally it wouldn't, unless the place you rented to consistently made reportings to a credit company (this is assume the eviction was for failure to pay). If the eviction was just for being a bad tenant (like "too noisy") it would never be reported to a credit agency. If the police were never involved, particularly if there was no conviction, there'd be no public police record either, most likely.
If you rented from a small agency or landlord, I'd bet dollars to donuts it's not there. Most landlords and rental companies have better things to do than put stuff on credit reports.
THAT SAID, there are some groups they can subscribe to - in certain states - that share information about bad tenants. Most don't, but some of the larger angencies might. At any rate, either they say yes or they say no. They won't throw you in jail, and a rejection wouldn't be on a credit report.
Most likely, you're okay. Just don't sign up for anything you can't handle, is all, unless you want to go through all that again.
Good luck!