Bought a home in 2013, lived there until June 2021, then rented it. If sold in June 2025, does the 2 out of 5 year rule apply to avoid tax on $240K gain? Moved for job.

The exclusion is 250K. 
You're eligible for the exclusion if you have owned and used your home as your main home for a period aggregating at least two years out of the five years prior to its date of sale. You can meet the ownership and use tests during different 2-year periods. However, you must meet both tests during the 5-year period ending on the date of the sale. 

Since your testing period started in 2020 you only lived in it for one of the last 5 years. But if you had to sell because of a job move more than 50 miles away you are eligible for a partial exclusion.