Question

what design pattern should be used to create own Object Container in C#, purpose is store and retrieve any type of objects, string, etc

Answer 1

Sounds like you are looking for a cache, you could create a wrapper around a Dictionary<TKey, TValue> e.g.

public class ObjectContainer
{
    private Dictionary<string, object> cache;

    public ObjectContainer()
    {
        cache = new Dictionary<string, object>();
    }

    public T Fetch<T>(string name)
    {
        object value;
        cache.TryGetValue(name, out value);
        return (T)value;
    }

    public void Add(string name, object value)
    {
        cache.Add(name, value);
    }
}
...

var cache = new ObjectContainer();
cache.Add("data1", 12345);
cache.Add("data2", new ClassName());
var intData = cache.Fetch<int>("data1");
var objData = cache.Fetch<ClassName>("data2");