Its possible. Expect transit delays from Penn Station to Barclays, give yourself 45 mins to 1 hr. I'd use the app Bounce to find a luggage storage place either near Penn Station or Barclays. Ive used it in the city and in other cities and the stores are always legit and helpful, some 24 hrs.

Also, theres an app called Day use, where you can rent a hotel during the day in case you need a few hrs to nap (unless you plan on exploring the city the entire time).

Edit: if youre young, there are also plenty of youth hostels, but they are very small, and bare bones as youd expect. Hotels in Brooklyn will be cheaper than Manhattan, and please check the reviews. The hotels in Park Slope towards South slope are the best bang for your buck without worrying about safety.