Real PO Interview Question : Stories that can not fit in the Sprint, how will you handle?

Unless something changes that causes the team velocity to increase or the scope of the work to change, you are not likely to deliver all of that functionality in four Sprints. However, this is not set in stone.

As a Product Owner, your role is to maximize the delivered value. The first step of that would be to work with the Development Team to order the work such that the most valuable work is delivered as quickly as possible, with any prerequisite work done first. Once the work has been ordered, the team simply works on the backlog, from top to bottom. Over time, the contents and ordering may change, but dependencies need to be minded. After four Sprints, you will have delivered as much value as the team is capable of delivering.

It's perfectly fine that you can't add additional resources. In fact, adding additional people will likely slow the team down as the new people are ramped up. Significant process changes may also have a negative impact as the team adjusts to them, so care must be taken in what experiments are run. If you're under pressure, the team's Scrum Master should be ensuring that impediments are identified and removed as quickly as possible to maximize the success of the team. At the same time, the Scrum Master needs to be advocating to ensure the team doesn't get put into a death march / crunch mode style project.

I'd also argue that if you have a velocity of 25, the 34 point Product Backlog Item needs to be decomposed. Perhaps even the 21 point backlog items. Typically, a backlog item is something that should fit into a Sprint. These would occupy most of the team's capacity for a Sprint. Without knowing what these items are, I can't say much, but I'm almost positive that there are ways to try to deliver some parts of the functionality earlier, obtain feedback, and reduce not only the size and complexity, but also the risk.

I'd also say that the Product Owner jumping in as a member of the Development Team is not a good solution. The Product Owner has a lot of work to do in order to maintain the Product Backlog. Especially in a case like this where it's not likely that the entire scope of work can be delivered in the desired timeframe. The goal of the Product Owner needs to be figuring out how to maximize the value delivered and managing expectations. It's definitely a full-time job.

In my opinion your answer is definitely one option. As a former Developer and QA, I have pitched in to help my teams when they requested it. But I always wait for them to ask or raise an impediment so that I am not undermining their ability to self-manage.

Second option.  This quote from the Scrum Guide section describing the Development Team. 

• They are self-organizing. No one (not even the Scrum Master) tells the Development Team how to turn Product Backlog into Increments of potentially releasable functionality;

As PO you can not dictate that all of the work be done by a specific date.  You can do your best to influence this by ordering the backlog in a manner that will allow the team to deliver the value incrementally. So in reality the question you were asked is invalid given that Scrum is focused on delivering incremental value to the point of satisfying the needs instead of delivering a set scope of work in a set period of time. 

A third option, similar to the second, would be that as PO it is your responsibility to maximize the value that they Development Team delivers. In addition the organization should honor your decisions knowing that you are not going to intentionally do anything that will be detrimental.  So, as a PO I would order the work needed in a manner that the Development Team delivered as much value as possible incrementally in the shortest possible time.  It may mean that some of the requested features might not be delivered.  But remember that Scrum isn't about delivering to a deadline. It is about delivering incremental value to the Stakeholders on a consistent basis. It is quite possible that after delivering a portion of that 34, it could be determined that whole thing isn't really needed. If a firm deadline is presented you have to compromise on something else. 

I'm sure that others will have a better answer and honestly I can't wait to read them.  

If Scrum is to be usefully applied, at least some valuable functionality of release quality must be delivered each and every Sprint, no matter how small.

Could that affect the strategy taken?

Team Velocity =. 25

I have items of following values how could I fit them in

34, 21, 21, 13, 13, 8, 5, 5, 3, 3

I can not break the 34 point task any further and I need to deliver this in 4 sprints without any additional resources.. How will I deliver? 

>> As far as only 'forecasting' is concerned, it looks tough on paper to deliver those in those 4 Sprints keeping everything constant (Scope,cost,time,quality).

But as other mentioned already Scrum is about delivering usable increment each sprint as per the Team's DOD and not meeting a deadline. Scope is volatile in scrum. Since Scrum is based on empiricism it can also happen that after delivering the first few increments the value of those PBI's might decrease or most part of the highest valued item is descoped. Even the velocity for that matter cant be constant. It can also happen in reality that you end up delivering all of them sooner. 

As rightly mentioned by @Daniel -  If a firm deadline is presented you have to compromise on something else. 

What's the right answer for this situation?

To answer your last question - As per me, I would take the highest ordered items and would target to deliver a useable increment each sprint. More will known after that and i might have to change my order , my strategy , values , efforts. 

"I've never seen a story of 34 points that could not be broken down further; I think we should look deeper into the 'why' behind not being able to break it down. Also, we would need to determine the items that bring the least value to the customer and put those on a later release, unless by some miracle we can finish the higher valued items early and then we can start working on those. Lastly, why is the release date 'hard-coded' for 4 sprints from now?"

Truly, the "right" answer depends on the person running the interview. Some people will expect a specific answer and even if your suggestion would work; they will still see it wrong.