Geauga County, southeast of Cleveland, is accepting Housing Choice Voucher Program (Section 8) applications, but apparently has a waiting list:
http://www.co.geauga.oh.us/departments/metro_housing.htm
It also appears that Lorain County, west of Cleveland, is accepting Section 8 applications:
http://www.lmha.org/section8/gosection8.htm
Some housing authorities have a residency requirement, so you need to check with the individual housing authority:
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http://www.cmha.net/hcvp/waitinglistinformation.aspx
Lake County, east of Cleveland, is seeking applicants for public housing units:
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While Lake County is not accepting Housing Choice Voucher Program (section 8) applications currently, you might want to check the public housing options as some of the Lake County public housing units aren't bad.
http://www.lakehousing.org/
Other northeast Ohio housing authorities include Akron and Ashtabula.
http://www.hud.gov/offices/pih/pha/contacts/states/oh.cfm
BOL!