Is it possible to use Session in Windows Service?

You can store in an object. The "Service" is an instance of your class, so you can store anything inside it.

public class MyTask
{
    public void LongOperation()
    {
        // seu código que pode demorar aqui
    }

    public Task LongOperationAsync()
    {
        return Task.Run(delegate () { this.LongOperation(); });
    }
}

public class MyService : ServiceBase
{
    Timer _timer;
    MyTask _currentTask;
    bool _called;

    protected override void OnStart(string[] args)
    {
        base.OnStart(args);
        _currentTask = new MyTask();
        _called = false;
        _timer = new Timer(5000);
        _timer.Elapsed += TimerElapsed;
        _timer.Start();
    }

    private async void TimerElapsed(object sender, ElapsedEventArgs e)
    {
        // TODO:
        //      talvez você precise usar uma sincronização à nível de Thread (lock) aqui
        if (_called)
        {
            // função ja foi chamada
            return;
        }
        _called = true;
        await _currentTask.LongOperationAsync();
        _called = false;
    }

    protected override void OnStop()
    {
        _timer.Stop();
        base.OnStop();
    }
}