Is it possible to create stateful web service in C#?

Stateful web services are not scalable and I wouldn't recommend them. Instead you could store the results of expensive operations in the cache. This cache could be distributed through custom providers for better scalability:

[WebMethod]
public string Method1()
{
    SomeObj so = TryGetFromCacheOrStore<SomeObj>(() => SomeClass.GetSomeObj(), "so");
    return so.Method1(); //this exetus in a moment 
}

[WebMethod]
public string Method2()
{
    SomeObj so = TryGetFromCacheOrStore<SomeObj>(() => SomeClass.GetSomeObj(), "so");
    return so.Method2(); //this exetus in a moment 
}

private T TryGetFromCacheOrStore<T>(Func<T> action, string id)
{
    var cache = Context.Cache;
    T result = (T)cache[id];
    if (result == null)
    {
        result = action();
        cache[id] = result;
    }
    return result;
}