Food Not Bombs
One great place to go in any city in order to feed yourself is the local Food Not Bombs.
Cleveland does have a Food Not Bombs branch, and they are very active. Every Saturday, they have a Weekly FoodShare at the intersection of Lorain Avenue and West 25th street in the "Market Square" Park. This typically happens at 5 PM, although, to stay updated, you should visit the website. Cleveland Food Not Bombs
If you want to get involved with this Weekly FoodShare, their cooking location is at 4241 Lorain Avenue, Cleveland, OH. It's just up the street. They meet at noon, and they have free coffee and baked goods if you show up. Not a bad deal, actually. Not only do you get to help the needy, you also get some quick food in exchange for your work.
The West Side Catholic Center
Just down Lorain Road from Market Square, at 3135 Lorain Avenue, Cleveland has a local organization called the West Side Catholic Center. The West Side Catholic Center is a veritable oasis in the desert that is homelessness. They offer nearly anything a homeless person could need. They offer clothing, shelter, emergency services, advocacy, and, most importantly, they offer hot food. They also offer some household goods and shelter for women and children. The West Side Catholic center is a very personal place, and you can't beat it when it comes to all around care.
Be warned, however. This place is often very crowded, and it has a large client base. Be prepared for long waits and nights that may be less successful than you want them to be. This place is, for all intents and purposes, a soup kitchen. And poverty in Cleveland is at an all-time high. There are a lot of people in need there. West Side Catholic Center