I have income tax in box 19 of my W2, is this school district tax? I can't find the Barnesville School district code in the list.

Barnesville SD code is 0701. It is non taxing district. The code in Box 20 is most likely for a City, not a school district, if there is withholding shown in box 19. 

To find out if you live in a school district or city with an income tax enter your address at:

http://www.tax.ohio.gov/Individual/LocalTaxInformation.aspx

Direct to ZIP + 4: https://thefinder.tax.ohio.gov/StreamlineSalesTaxWeb/ZipLookup/LookupByZip.aspx?taxType=taxsummary 

To see if your work location is in a city with a tax, enter your work address instead.

 You normally only have to file a return in the city you live in. It is not necessary to file one in the city you work in, if different. Your employer will remit withholding to the city you work in, but you do not file a return unless you're claiming a refund or your employer didn't withhold enough ( both highly unusual). The city you live in will allow a credit or partial credit for the withheld tax you paid to the work location city. 

Turbotax cannot prepare a Barnesville city return (Barnesville is neither a RITA or CCA member).

City Income tax division web site http://www.barnesvilleohio.com/fiscal.htm