Business can be used as either a countable noun or an uncountable noun.
Your examples should be written,
It goes into business as well.
He is running a business.
As I said once in another answer of mineanother answer of mine, most dictionaries give the definitions without saying anything about a noun being countable or uncountable. However, some dictionaries do. To check the countability of a noun online, I recommend Macmillan Dictionary and Collins Dictionaries.
You can consult, for example, Macmillan about the word business.
The business in "It goes into business as well." is defined in sense 1,
1 [uncountable] the work of buying or selling products or services for money
representatives from business and politics
do business with someone: I found them very easy to do business with.
go into business: It was a mistake to go into business with my brother.
The business in "He is running a business." is defined in sense 2,
2 [countable] an organization that buys or sells products or services for money. This word often refers to a small organization that is owned and run by one person or a few people, while company can also refer to large organizations
a small family business
Matthew's little shop has turned into a thriving business.
Many small businesses fold (=fail) in their first year.
run a business: Sheryl's parents run a small clothing business.
start your own business: After leaving school, Bob started his own computer business.
There are many other example sentences that could help you understand the different senses of words in both of the two dictionaries.